Download Modeling and Design of Photoconductive and Superconductive

Modeling and Design of Photoconductive and
Superconductive Terahertz Photomixer Sources
by
Daryoosh Saeedkia
A thesis
presented to the University of Waterloo
in fulfilment of the
thesis requirement for the degree of
Doctor of Philosophy
in
Electrical and Computer Engineering
Waterloo, Ontario, Canada, 2005
c
°
Daryoosh Saeedkia 2005
I hereby declare that I am the sole author of this thesis. This is a true copy of
the thesis, including any required final revisions, as accepted by my examiners.
I understand that my thesis may be made electronically available to the public.
ii
Abstract
Terahertz technology is a fast growing field with variety of applications in biology
and medicine, medical imaging, material spectroscopy and sensing, monitoring and
spectroscopy in pharmaceutical industry, security, and high-data-rate short-range
communications. Among different terahertz sources, photomixers are potentially
compact, low power consuming, coherent, low-cost, and tunable continuous-wave
sources. A terahertz photomixer is a heterodyne scheme, in which two laser beams
with their frequency difference falling in the terahertz range mix in a nonlinear
medium, such as a photoconductor or a superconductor, and generate a signal,
whose frequency is equal to the frequency difference of the two lasers. The frequency
of the generated terahertz signal can be tuned by tuning the central frequency of
one of the lasers.
In this dissertation, the photomixing in superconductors and photoconductors is
studied, and comprehensive analytical models for the interaction of two interfering
laser beams with these materials are developed. Integrated photomixer-antenna
elements as efficient terahertz sources are introduced and arrays of these elements
as high power terahertz sources are designed. Also, an array of photoconductive
photomixer-antenna elements with integrated excitation scheme is proposed.
In a photo-excited superconductor, the fundamental equations for the motion of
the carriers inside the superconductor material are used in connection with the
two-temperature model to find an analytic expression for the generated terahertz
photocurrent inside the film. In a photo-excited photoconductor, the continuity
equations for the electron and hole densities are solved in their general form along
with the appropriate boundary conditions to find photocurrent distribution inside
the photoconductor film. It is shown that in a continuous-wave (CW) terahertz
photomixing scheme, the resulting photocurrent contains a dc component and a
iii
terahertz traveling-wave component. The dependency of the amplitude and the
phase of the generated photocurrent on the physical parameters of the photomixer,
the parameters of the lasers, the applied dc bias, and the configuration of the device
is explored in detail for a photoconductive photomixer made of low-temperature
grown (LTG) GaAs and for a high-temperature superconductive photomixer made
of YBa2 Cu3 O7−δ .
The developed models for the photoconductive and the superconductive terahertz
photomixers are used to design new integrated photomixer-antenna devices. In
these devices, the photomixing film simultaneously acts as an efficient radiator at
the terahertz frequencies. Integrating the photomixing medium with the antenna
not only eliminates any source to antenna coupling problem, but also makes the
proposed device attractive for array configurations.
To increase the generated terahertz power, arrays of the photoconductive and
the superconductive photomixer-antenna elements are proposed as CW terahertz
sources. It is shown that a sub-milliwatt terahertz power is achievable from a
typical superconductive photomixer-antenna array structure. The beam steering
capability of the proposed devices is also investigated.
A photoconductive photomixer-antenna array with integrated excitation scheme is
proposed, in which the laser beams are guided inside the substrate and excite the
photomixer elements. In this way the laser power is only being consumed by the
photomixer elements, and the photomixer-antenna elements can be integrated with
other optical components on a single substrate. The whole structure is robust and
less sensitive to vibration and other environmental parameters.
iv
Acknowledgments
I would like to express my sincere thanks to Professor Safieddin Safavi-Naeini and
Professor Raafat R. Mansour for their supervision and support during my Ph.D.
program. During this journey, I learned many things that I would not be able
to learn without their help and I am greatly thankful to them for giving me this
opportunity.
I would like to thank to my thesis defense committee members, Professor Frank
Hegmann from Department of Physics, University of Alberta, Edmonton, Canada,
Professor James Martin from Department of Physics, University of Waterloo, Professor Sujeet Chaudhuri from Department of Electrical and Computer Engineering,
University of Waterloo, and Professor Amir keyvan Khandani from Department
of Electrical and Computer Engineering, University of Waterloo, for reading my
thesis, for their invaluable comments and suggestions, and for accepting my thesis
as a Ph.D. dissertation.
During these years, my darling wife Nadia was the one who was always there for
me. She carried out all the responsibilities of our common life and gave me the
chance to concentrate on my work. I could not have done this without her support.
Words cannot express my appreciation for what she has done to me.
I would like to thank to Wendy Boles, Graduate Studies Coordinator, and Wendy
Gauthier, former Graduate Office Admissions Secretary, at the Department of Electrical and Computer Engineering, for their help and guidance all these years.
It was a great opportunity for me to pursue my Ph.D. program at one of the best
universities in Canada. With its warm and friendly atmosphere, the University of
Waterloo is a great place for international students to continue their education. I
will never forget all the good things that happened to me in this university during
the last three years.
v
To my darling wife Nadia and my parents
vi
Contents
1 Introduction
1
1.1
Motivation and Summary of Work . . . . . . . . . . . . . . . . . . .
1
1.2
Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . .
4
2 Continuous-Wave Terahertz Sources: A Technology Overview
6
2.1
Electron Beam Sources . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.2
Optically Pumped Far-Infrared Gas Lasers . . . . . . . . . . . . . .
8
2.3
Solid-State Sources . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.4
Frequency Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.5
Terahertz Semiconductor Lasers . . . . . . . . . . . . . . . . . . . .
11
2.6
Terahertz Parametric Oscillators . . . . . . . . . . . . . . . . . . . .
12
2.7
Terahertz Photomixers . . . . . . . . . . . . . . . . . . . . . . . . .
15
3 Photomixing in Superconductors
21
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.2
Modeling of Superconductive Photomixers . . . . . . . . . . . . . .
22
vii
3.3
Simulation Results for an YBCO Photomixer
. . . . . . . . . . . .
4 Photomixing in Photoconductors
33
49
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.2
Modeling of Photoconductive Photomixers . . . . . . . . . . . . . .
51
4.3
Simulation Results for a Photomixer made of LTG-GaAs . . . . . .
62
5 Integrated Photomixer-Antenna Elements
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Longitudinal and Transversal PhotomixerAntenna Elements
77
5.2.1
Radiation from Photomixer-Antenna Elements . . . . . . . .
79
5.2.2
Simulation Results for an LTG-GaAs Photomixer-Antenna
6 Photomixer-Antenna Array
81
88
Photoconductive Photomixer-Antenna Array . . . . . . . . . . . . .
88
6.1.1
Array Design Procedure . . . . . . . . . . . . . . . . . . . .
93
6.1.2
Simulation Results for an LTG-GaAs Photomixer-Antenna
Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2
76
. . . . . . . . . . . . . . . . . . . . . . . . . . .
Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1
76
95
Superconductive Photomixer-Antenna Array . . . . . . . . . . . . . 106
6.2.1
Array Design Procedure . . . . . . . . . . . . . . . . . . . . 109
6.2.2
Simulation Results for an YBCO Photomixer-Antenna Array 110
viii
7 Photomixer-Antenna Array with Integrated Excitation Scheme 118
7.1
Array Design I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.2
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.3
Array Design II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.4
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
8 Concluding Remarks
134
8.1
Summary and Contributions . . . . . . . . . . . . . . . . . . . . . . 134
8.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A Calculation of the Carrier Densities N0 and P0
ix
138
List of Figures
2.1
(a) Schematic of a terahertz parametric oscillator (TPO)(b) Injectionseeded terahertz parametric generator (IS-TPG).
2.2
. . . . . . . . . .
13
(a) Top view of interdigitated-electrode vertically-driven photomixer
coupled to a planar dipole antenna (b) Cross-sectional view of photomixer showing bottom-side coupling of THz radiation through a
dielectric lens to free space. . . . . . . . . . . . . . . . . . . . . . .
2.3
16
Equivalent circuit diagram for the photomixer and antenna. Current drawn from the bias source is modulated at angular frequency
ω by the photoconductor, with resistance R(ω, t). The capacitance
C represents the effects of the charge accumulating at the electrodes.
Power is dissipated in the antenna, RA (ω), of which the component
oscillating with angular frequency ω is coupled out as terahertz radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
17
(a) Velocity-match of laser interference fringes to the THz wave on
the coplanar stripline is obtained by a tuning-angle between the two
laser beams (b) Scanning electron microscope picture of a travelingwave metal-semiconductor-metal (TW-MSM) structure with inner
part of bow-tie antenna (c) A section of TW-MSM. . . . . . . . . .
x
18
2.5
Output power of different continuous-wave terahertz sources versus
frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.1
A typical superconductive photomixer scheme. . . . . . . . . . . . .
23
3.2
Critical optical power versus distance from the surface of the superconductor film at different bath temperatures. The critical temperature is Tc = 87 K. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
Amplitude of the time-varying components of the electron and phonon
temperatures. The dc current bias is 23.8 mA. . . . . . . . . . . . .
3.4
38
Amplitude of the time-varying component of the electron temperature versus bath temperature at different beat frequencies. . . . . .
3.5
35
39
Drift and diffusion components of photocurrent given at the right
hand side of Eq. (3.41) for two different bath temperatures. The
critical temperature is Tc = 87 K. . . . . . . . . . . . . . . . . . . .
3.6
40
Total photocurrent at the surface of the YBCO film versus beat frequency for θ = 10o and at different bath temperatures. The critical
temperature is Tc = 87 K. . . . . . . . . . . . . . . . . . . . . . . .
3.7
41
Total photocurrent at the surface of the YBCO film versus beat
frequency for θ = 0 and at different bath temperatures. The critical
temperature is Tc = 87 K. . . . . . . . . . . . . . . . . . . . . . . .
3.8
Total photocurrent at the surface of the YBCO film versus bath
temperature for θ = 10o and at different beat frequencies. . . . . . .
3.9
42
43
Total photocurrent along the thickness of the YBCO film at T0 = 77K. 45
3.10 Total photocurrent along the thickness of the YBCO film at T0 = 82K. 46
xi
3.11 Phase of photocurrent versus angle between the two laser beams. . .
47
3.12 Generated terahertz power inside the YBCO film versus total incident optical power at two different bath temperatures. The applied
dc bias is 23.8 mA . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4.1
A typical photoconductive photomixer scheme. . . . . . . . . . . . .
52
4.2
Electron lifetime versus applied electric field in LTG-GaAs. The
low-field electron lifetime is τn0 = 0.1 ps. . . . . . . . . . . . . . . .
4.3
64
Amplitude of electron density inside the LTG-GaAs film versus electron lifetime at different beat frequencies. The total applied optical
power density is I0 = 0.4 mW/µm2 . . . . . . . . . . . . . . . . . . .
4.4
65
Amplitude of electron density inside the LTG-GaAs film versus applied electric field at different beat frequencies. The total applied
optical power density is I0 = 0.4 mW/µm2 . . . . . . . . . . . . . . .
4.5
67
Amplitude of electron density versus beat frequency for different electron lifetimes in an LTG-GaAs photomixer. The total applied optical
power density is I0 = 0.4 mW/µm2 . . . . . . . . . . . . . . . . . . .
4.6
68
Amplitude of terahertz equivalent surface photocurrent in LTG-GaAs
photomixer versus applied electric field at different beat frequencies
and for two different low-field electron lifetimes. The total applied
optical power density is I0 = 0.4 mW/µm2 . . . . . . . . . . . . . . .
4.7
69
Amplitude of dc photocurrent in LTG-GaAs photomixer versus applied electric field at different low-field electron lifetimes. The total
applied optical power density is I0 = 0.4 mW/µm2 . . . . . . . . . .
xii
71
4.8
Amplitude of terahertz equivalent surface photocurrent in LTG-GaAs
photomixer versus beat frequency at different low-field electron lifetimes. The total applied optical power density is I0 = 0.4 mW/µm2 .
4.9
72
Amplitude of terahertz equivalent surface photocurrent versus thickness of the LTG-GaAs film for different absorption coefficients. . . .
74
4.10 Longitudinal component of grating vector, Kx , and phase of terahertz photocurrent, φ, versus angle between the two laser beams. . .
75
5.1
Longitudinal photoconductive photomixer scheme. . . . . . . . . . .
77
5.2
Transversal photoconductive photomixer scheme. . . . . . . . . . .
78
5.3
Amplitude of equivalent surface photocurrent in LTG-GaAs longitudinal and transversal photomixers. The total optical power density
of the two lasers is 0.4 mW/µm2 . The applied dc bias to each element is 4 V/µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4
82
Radiation power of LTG-GaAs longitudinal and transversal photomixers. The total power density of the two lasers is 0.4 mW/µm2 .
For the longitudinal case, the length of the film is l = 5.6 µm, while
the width of the film is w = 50 µm. For the transversal case, the
length and the width of the film are l = 50 µm and w = 5.6 µm, respectively. The thickness of the film for both cases is d = 2 µm. The
thickness of the substrate is changed according to the beat frequency
to have maximum radiation at each frequency. The applied dc bias
to each element is 4 V/µm.
5.5
. . . . . . . . . . . . . . . . . . . . . .
84
Configuration factor, ξ, for longitudinal and transversal photomixerantenna devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
85
5.6
Radiation power of LTG-GaAs longitudinal photomixer designed for
2 THz beat frequency. The length of the film is l = 5.6 µm and the
width of the film is w = 50 µm. The thickness of the substrate is
fixed at h = 10.917 µm to have maximum radiation at 2 THz. The
applied dc bias to each element is 4 V/µm. . . . . . . . . . . . . . .
6.1
87
(a) Photoconductive photomixer-antenna array configuration (b) Optical excitation scheme. . . . . . . . . . . . . . . . . . . . . . . . . .
89
6.2
Array arrangement in (x − y) plane. . . . . . . . . . . . . . . . . . .
91
6.3
Total terahertz radiation power of an 1×31 photoconductive photomixerantenna array made of LTG-GaAs versus beat frequency. The thickness of the substrate is fixed at 21.8 µm to have maximum radiation at 1 THz. The total optical power density of the two lasers is
0.12 mW/µm2 . The lengths of the array elements and their separation distances are calculated from Eqs. (6.17) to (6.19), and are
around 7.46 µm. The width of the array elements is 120 µm. The
elements are biased to a dc voltage of 7.5 V. . . . . . . . . . . . . .
6.4
96
Total terahertz radiation power of an 1×31 photoconductive photomixerantenna array made of LTG-GaAs versus applied electric field at different beat frequencies. The thickness of the substrate is changed
according to the beat frequency to have maximum radiation at each
frequency. The total power density of the two lasers is 0.12 mW/µm2 . 97
xiv
6.5
Total terahertz radiation power of an 1×31 photoconductive photomixerantenna array made of LTG-GaAs versus beat frequency. The thickness of the substrate changes according to the beat frequency to
have maximum radiation at each frequency. The total optical power
density of the two lasers is 0.12 mW/µm2 . . . . . . . . . . . . . . .
6.6
99
Radiation pattern at φ = 0 plane for different number of array
elements and at the beat frequency of 1 THz; (a) 1 × 31 array
(FWHM=6.59o ), (b) 1 × 51 array (FWHM=4o ), (c) 1 × 71 array
(FWHM=2.88o ), (d) 1 × 91 array (FWHM=2.3o ). . . . . . . . . . . 100
6.7
Radiation pattern at φ = 0 and φ = π/2 planes for an 1 × 31 array
at the beat frequency of 1 THz and θl = 3.02o . . . . . . . . . . . . . 102
6.8
Terahertz radiation power versus angle between the two lasers at the
beat frequency of 1 THz. . . . . . . . . . . . . . . . . . . . . . . . . 103
6.9
Terahertz radiation power versus number of array elements at the
beat frequency of 1 THz and for a constant optical power density of
0.12 mW/µm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.10 Terahertz radiation power versus total optical power density at the
beat frequency of 1 THz. . . . . . . . . . . . . . . . . . . . . . . . . 105
6.11 Superconductive photomixer-antenna array configuration. . . . . . . 106
6.12 Lengths of the array elements and their separation distances. . . . . 112
6.13 Terahertz radiation power versus angle between the two lasers. . . . 113
xv
6.14 Terahertz radiation power versus beat frequency for an YBCO array
with 61 elements designed to operate at 3 THz. The thickness of the
substrate is 5.21 µm. The thickness and the width of the array elements are 130 nm and 40 µm, respectively. The bath temperature is
T0 = 85 K. Each laser radiates 114 mW optical power. The applied
dc bias is 23.8 mA. . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.15 Total terahertz radiation power for an YBCO array with 61 elements versus beat frequency at two bath temperatures. The critical
temperature is Tc = 87 K. The thickness of the substrate changes
according to the beat frequency to have maximum radiation at each
frequency. The applied dc bias is 23.8 mA. . . . . . . . . . . . . . . 116
6.16 Radiation pattern of the array at φ = 0 plane for different angles
between the two lasers. . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.1
(a) A photomixer-antenna array with integrated excitation scheme.
The laser beams are guided inside the core layer made of Al0.1 Ga0.9 As
with refractive index of nf = 3.55 and excite the photomixer-antenna
elements made of LTG-GaAs. (b) Electric field distribution around
i-th element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
xvi
7.2
Terahertz radiation power versus beat frequency for different values
of the thickness of the photomixer-antenna elements. The number of array elements is 19.
The thickness of the core layer is
hf = 0.573 µm. The thickness of the substrate at the frequency of
1 THz is , hs = 22.4 µm. The length of the array elements at the
frequency of 1 THz is 55.6 nm. The width of the array elements
is 100 µm. The dc bias voltage over each element is 0.139 V. The
optical power coupled to the waveguide from each of the lasers is 100
mW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.3
Absorbed optical power in each element. . . . . . . . . . . . . . . . 128
7.4
A photomixer-antenna array with integrated excitation scheme: two
laser beams are lunched from one side of the optical waveguide. . . 129
7.5
Power flow diagram along the array structure. . . . . . . . . . . . . 132
7.6
Terahertz radiation power versus beat frequency for an array with
5 elements. The thickness of the core layer is hf = 0.573 µm. The
thickness of the substrate at the frequency of 1 THz is , hs = 22.4 µm.
The optical power of each laser is 100 mW. The lengths of the array
elements are 220 nm, 282 nm, 395 nm, 663 nm, and 2.814 µm. The
thickness and the width of the array elements are 0.573 µm and 200
µm, respectively. The absorbed optical in each element is 39.5 mW.
The applied electric field over each element is 2.5 V/µm. . . . . . . 133
xvii
List of Tables
2.1
Power and frequency range of some two-terminal solid-state devices
9
2.2
Performance of planar Schottky diode multipliers . . . . . . . . . .
10
2.3
Characteristics of state-of-the-art continuous-wave quantum cascade
lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.4
Characteristics of state-of-the-art terahertz parametric oscillators .
14
3.1
Physical parameters of a photomixer made of YBa2 Cu3 O7−δ . . . .
34
4.1
Physical parameters of a photomixer made of LTG-GaAs . . . . . .
63
xviii
Chapter 1
Introduction
1.1
Motivation and Summary of Work
The terahertz spectrum, or far-infrared, refers to the frequency range from 100 GHz
up to a few 10 THz. Terahertz technology is a fast growing field with its variety of
new applications, including biology and medicine [1]- [6], imaging [7]- [9], material
spectroscopy and sensing [10]- [16], monitoring and spectroscopy in pharmaceutical
industry [17], [18], security [19]- [22], and high-data-rate communications [23]- [26].
Many chemical and biological matters have their own signatures in the terahertz
spectrum, which makes it possible to identify and analyze them based on the way
that they interact with terahertz waves.
Unlike X-rays, terahertz radiation is non-ionizing and can be safely used in medical
imaging systems [27], [28]. Terahertz medical imaging presents unique solutions for
a variety of health related problems such as tissue identification through its water
content [29], disease diagnostics such as skin and liver cancer detection [30]- [33],
and identification of dental caries [34], [35].
1
Terahertz waves can penetrate many non-metallic enclosures and can be used for
screening and detecting hidden objects such as weapons containing a small amount
of metal, ceramic weapons, explosive materials, and chemical and biological threats
[19]- [21].
By increasing the number of users and services in wireless communication industry,
moving to higher carrier frequencies to have more bandwidth for communication
and data transferral is inevitable. Although the high atmospheric attenuation at
terahertz frequencies makes it difficult to have a long-range mobile communication,
however a high-bandwidth, short-range, and line of sight wireless link is completely
realizable. The atmospheric attenuation is an advantage to reduce the coverage
range of the signal in military applications to avoid communication being overheard,
or in frequency re-use applications to avoid signal interfering.
Among different continuous-wave terahertz sources, photomixers are one of the most
promising ones as potentially compact, low power consuming, low-cost, coherent,
and tunable terahertz sources [36]- [65]. A terahertz photomixer is a heterodyne
scheme, in which outputs of two single-mode lasers or output modes of a dualmode laser with their frequency difference falling in the terahertz range mix in a
nonlinear medium, such as a photoconductor or a superconductor, and generate a
signal, whose frequency is equal to the frequency difference of the two lasers or two
modes of the dual-mode laser. The frequency of the generated terahertz signal can
be tuned by tuning the central frequency of the lasers. In fact, photomixers take
advantage of both microwave and photonic techniques to generate signals in the
terahertz frequency range.
Narrow spectral bandwidth and frequency control of the photomixers make them
suitable for the applications such as biological and material spectroscopy [66]- [69],
medical imaging [70], [71], and security [72], which need a tunable source over the
2
terahertz spectrum. Photomixers also can be used as local oscillators in heterodyne
receivers for radio astronomy and communication applications [53].
The main drawback of terahertz photomixers is their low output terahertz power
coming from their low optical-to-electrical conversion efficiency [73]. Much research
has been conducted to increase the available power from these sources [74]- [78].
As a fundamental research work on the terahertz photomixers, we have developed
analytical models for the interaction between laser beams and photoconductors and
superconductors in typical photomixing schemes [79], [80].
Based on these models, we have proposed integrated photomixer-antenna elements
made of low-temperature-grown (LTG) GaAs or YBa2 Cu3 O7−δ high-temperature
superconductor [81], [82]. Unlike conventional photomixers, in which a generated
terahertz signal in an active region is coupled to an antenna for radiation, in a
photomixer-antenna element the generated terahertz signal inside the photomixing
film radiates simultaneously by designing the film as an efficient radiator. Using
a photomixing medium as an antenna element not only eliminates any source-toantenna coupling problem, but also distributes the optical power over a bigger area
and hence increases optical power handling capability of the device.
We have designed arrays of the photomixer-antenna elements that generate more
terahertz power and have beam-steering capability [83]- [85]. We have also proposed
an array structure with an integrated optical excitation scheme. In this structure
the photomixer elements are located above a dielectric substrate, which at the
same time is a waveguide for the exciting laser beams. The laser beams excite the
photomixer elements while they travel along the waveguide. In this way the laser
power is only being consumed by photomixer elements, and hence the efficiency of
the source increases.
3
1.2
Organization of Thesis
Chapter 2 reviews the state-of-the-art technologies of continuous-wave terahertz
sources. Different CW terahertz source technologies are compared against each
other from their maximum available terahertz power point of view.
Photomixing in superconductors is discussed in Chapter 3. For a photo-excited
superconductor film the fundamental equations for the carriers motion inside the
superconductor material have been used in connection with the two-temperature
model to find an analytic expression for the generated terahertz photocurrent inside
the film. The dependency of the resulting photocurrent on the material and laser
parameters, temperature, and photomixer configuration for a photomixer made of
YBa2 Cu3 O7−δ high-temperature superconductor is studied.
Photomixing in photoconductors is discussed in Chapter 4. For a dc biased photoconductor excited by two laser beams the continuity equations for the electron and
hole densities are solved in their general form along with the appropriate boundary
conditions to find generated terahertz photocurrent distribution. The dependency
of the photocurrent on the applied bias, parameters of the lasers, and photomixer
configuration is explored for photomixers made of low-temperature-grown GaAs.
In Chapter 5, the integrated photomixer-antenna element is introduced. In this
element the generated terahertz photocurrent radiates simultaneously by designing
the photomixing film as an efficient radiator. The radiation characteristics for a
photoconductive photomixer-antenna element over a grounded dielectric substrate
are studied.
In Chapter 6, photoconductive and superconductive photomixer-antenna arrays are
introduced. In these configurations, arrays of the photomixer-antenna elements are
designed properly to radiate maximum power. The beam steering capability of
4
these configurations are also investigated.
The photoconductive photomixer-antenna array with integrated optical excitation
scheme is introduced in Chapter 7. In this structure, the laser beams are guided
inside the substrate and excite the photomixer elements while they travel along the
length of the waveguide. Two different array structures are designed, for which the
design procedures and the radiation characteristics are discussed in detail.
Chapter 8 contains some concluding remarks and provides suggestions for future
works based on what is done in this dissertation.
5
Chapter 2
Continuous-Wave Terahertz
Sources: A Technology Overview
Continuous-wave terahertz sources can be divided into seven categories, including
electron beam sources, optically pumped far-infrared gas lasers, solid-state sources,
frequency multipliers, terahertz semiconductor lasers, terahertz parametric oscillators, and photomixers. In the following, we review the existing technologies for
these sources.
2.1
Electron Beam Sources
Gyrotrons [86], free electron lasers (FEL) [87], [88], and backward wave oscillators
(BWO) [89] are electron beam sources that generate relatively high power signals
at the terahertz frequency range. All of these devices work based on the interaction
of a high-energy electron beam with a strong magnetic field inside resonant cavities
or waveguides. Due to this interaction, energy transfer occurs from the electron
6
beam to an electromagnetic wave.
In gyrotrons cavities and waveguides are operated in high-order modes to increase
the cross-sectional dimensions at high-frequency operations. Gyrotrons with 100
KW power operating in CW regime at 100 GHz [90] and with 1 MW power at
140 GHz [91] have been successfully developed. It has been shown that a gyrotron
with 2 MW power at 170 GHz is feasible [92]. Free electron lasers are basically
gyrotrons with very high operating frequencies obtained by deriving the electron
beam with a high energy electron accelerator, and employing an optical cavity [93].
They also have greater tunability compared with gyrotrons. FELs are capable
of operation over virtually the entire electromagnetic spectrum. They have been
built at wavelengths from microwaves through the ultraviolet [94]- [97] and can be
extended to X-rays [98]. A free electron laser at the University of California, Santa
Barbara works at far-infrared region and generates 1 KW quasi-continuous-wave
signal at 300 GHz.
In BWOs the phase and group velocity of the electromagnetic wave have opposite
directions. BWOs are tunable sources with the possibility to tune their frequency
electrically over a bandwidth of more than 50% and can generate up to 50 mW
of power at 300 GHz going down to a few mW at 1 THz [99]. The commercially
available systems provided by a Russian company (ISTOK) cover frequency range
from 177 GHz to 1100 GHz with 1-10 mW output power. Complete systems are
heavy and large and need high bias voltage and water cooling system [100], although
they are much more smaller than FELs and gyrotrons. Microfabrication techniques
are promising to reduce the size of these devices to make them suitable for terahertz
applications [101].
Electron beam devices are bulky and need extremely high fields as well as high
current densities, which are main disadvantages of these devices.
7
2.2
Optically Pumped Far-Infrared Gas Lasers
These terahertz sources consist a CO2 pump laser injected into a cavity filled with
a gas that lases to produce the terahertz signals [102], [103]. The lasing frequency
is fixed dictated by the filling gas. Tunable sources have been developed based
on mixing a tunable microwave source with these gas lasers [104], [105]. Power
levels of 1-20 mW are common for 20-100 W laser pump power depending on the
chosen line, with one of the strongest being that of methanol at 2522.78 GHz. A
miniaturized gas laser has been reported with 75 × 30 × 10 cm dimensions and 20
kg weight, which generates 30 mW power at 2.5 THz [106].
2.3
Solid-State Sources
Solid-state sources can be devided to two-terminal and three-terminal devices [107][109]. Two-terminal devices include resonant tunneling diodes (RTDs) [110], Gunn
or transferred electron devices (TEDs) [111]- [113], and transit time devices such as
impact avalanche transit time (IMPATT) diodes and tunnel injection transit time
(TUNNETT) diodes. All of these devices exhibit a negative differential resistance
property, although with different mechanisms.
The oscillation frequency of 578 GHz with GaInAs/AlAs RTD with 0.2 µW output
power [114] and 712 GHz with InAs/AlSb RTD with 0.3 µW output power [115]
have been obtained, which the latter is the highest fundamental frequency achieved
so far from a solid-state source. Recently a carbon nanotube RTD device has been
proposed as a CW terahertz source, which predicts 2.5 µW terahertz power at 16
THz [116]. The state-of-the-art Gunn devices generate 0.2 to 5 µW power at 400560 GHz frequency range [117]. Presently, the maximum operation frequency for a
8
Table 2.1: Power and frequency range of some two-terminal solid-state devices
Device
Power
Frequency
RTD (InAs/AlSb) [115]
0.3 µW
712 GHz
RTD (InGaAs/AlAs) [114], [118]
0.2 µW
50 µW
200 µW
578 GHz
205 GHz
100 GHz
Gunn (InP) [108], [111], [117], [119]
5 µW
2 mW
25 mW
80 mW
130 mW
416
315
162
152
132
IMPATT (Si) [120], [121]
0.2 mW
7.5 mW
300 mW
361 GHz
285 GHz
140 GHz
TUNNETT (GaAs) [119], [122]
0.14 mW
4 mW
9 mW
10 mW
355
235
210
202
GHz
GHz
GHz
GHz
GHz
GHz
GHz
GHz
GHz
TUNNETT diode is 355 GHz with 140 µW output power [122]. Table 2.1 shows the
available two-terminal solid-state devices with their frequency and power ranges.
Among three terminal solid-state devices, field effect transistors are suitable for
high frequency operations because of their submicrometer gate lengths. A high
electron mobility transistor (HEMT) with 30 nm gate length and 547 GHz cutoff
frequency has been developed, which is the highest cutoff frequency yet reported
for any transistor [123]. An InP HEMT monolithic millimeter-wave integrated
circuit (MMIC) amplifier has been designed, which generates 20 mW power at
150 GHz [124]. Three terminal devices have advantage over two terminal devices
9
Table 2.2: Performance of planar Schottky diode multipliers
Multiplier chain
×2(×3)2 [127]
×2(×3)2 [128]
×2(×3)2 [128]
(×2)4 [129]
(×2)4 [129]
(×2)2 × 3 [130]
(×2)2 × 3 [130]
Temperature Output power
120 K
273 K
120 K
273 K
120 K
273 K
80 K
2.6 µW
3 µW
15 µW
15 µW
40 µW
75 µW
250 µW
Frequency
1.9 THz
1.74 THz
1.74 THz
1.5 THz
1.5 THz
1.2 THz
1.2 THz
in low-noise preamplifier applications. A 220 GHz HEMT low-noise amplifier has
been developed with 10 dB gain over the bandwidth from 180 to 225 GHz [125].
2.4
Frequency Multipliers
In terahertz frequency multipliers, the frequency of a drive source is multiplied in a
nonlinear device to generate higher order harmonic frequencies. Schottky varactor
diodes are predominantly used as nonlinear devices in multipliers. Planar Schottky
varactor diodes are commonly used in frequency multipliers, taking the advantage
of GaAs substrateless technology to reduce substrate loss. The drive sources can
be BWOs as well as solid-state sources such as Gunn and IMPATT oscillators,
with relatively high output power in the range of 50 GHz to 150 GHz. Microwave
frequency synthesizers in combination with high-gain power amplifiers fabricated
by the monolithic microwave integrated circuit (MMIC) technology can provide
a high power source above 100 GHz [126]. The most efficient terahertz frequency
multipliers are realized by series chains of frequency doublers and frequency triplers
[131]. The performances of the state-of-the-art planar Schottky diode multipliers
10
are given in Table 2.2. To date, the maximum achievable frequency from the
frequency multipliers has been limited to 2 THz [132], [133].
2.5
Terahertz Semiconductor Lasers
In the terahertz semiconductor lasers, the inversion of population takes place between discrete energy levels, which generates terahertz stimulated emission. The
first terahertz semiconductor laser working at 4 to 5 K was a single crystal of pdoped Ge placed between two mirrors [134]. The terahertz frequency can be tuned
in the range of 1-4 THz.
The most promising terahertz semiconductor lasers are quantum cascade lasers
[135]- [139]. The idea of quantum cascade laser was introduced in 1971 [135], while
the first realization became possible in 1994 [136]. A quantum cascade laser is a
unipolar laser, in which the conduction band or the valence band is divided to few
intersubbands. The carrier transition occurs between these discrete energy levels
within the same band. The discrete energy levels are created in a semiconductor heterostructure containing several coupled quantum wells. The first terahertz
quantum cascade laser working at 50 K and generating 2 mW power at 4.4 THz
contained 728 quantum wells, where each quantum well contained two 1-4 nm thick
AlGaAs barriers and one 10-20 nm thick GaAs well [140]. A quantum cascade laser
working in pulse regime with 3.5 W peak power at 33 THz operating at 300 K has
been reported [141].
The first continuous-wave quantum cascade laser generated 25 µW at 4.4 THz working at 52 K [142]. Given in Table 2.3 are the characteristics of the state-of-the-art
continuous-wave quantum cascade lasers. As it can be seen from Table 2.3, the
room temperature operation has not been achieved for frequencies below 10 THz.
11
Table 2.3: Characteristics of state-of-the-art continuous-wave quantum cascade
lasers
Output power Temperature Frequency
1.34 W [143]
26 mW [143]
1.1 W [144]
640 mW [144]
162 mW [144]
446 mW [145]
30 mW [145]
17 mW [146]
3 mW [146]
4 mW [147]
1.8 mW [148]
0.4 mW [148]
80 K
313 K
200 K
295 K
298 K
293 K
333 K
292 K
312 K
48 K
10 K
78 K
70 THz
70 THz
50 THz
50 THz
47.6 THz
50 THz
50 THz
33 THz
33 THz
4.4 THz
3.2 THz
3.2 THz
To date, the highest reported temperature for a terahertz quantum cascade laser
working at 3.2 THz is 93 K [148].
2.6
Terahertz Parametric Oscillators
The terahertz wave generation via the resonance of lattice and molecular vibration
in the nonlinear optical crystals was predicted by Nishizawa in 1963 [149], [150].
In a terahertz parametric oscillator, the near infrared input (pump) photon stimulates a Stokes photon (idler) at a different frequency between the pump photon and
the vibrational mode. The pump and idler photons interact inside the nonlinear
optical crystal and generate a terahertz wave (signal), whose frequency is equal to
the frequency difference of the pump and idler. In this process, the momentum is
conserved, which indicates that the terahertz frequency can be tuned by controlling
12
(a)
(b)
Figure 2.1: (a) Schematic of a terahertz parametric oscillator (TPO)(b) Injectionseeded terahertz parametric generator (IS-TPG).
the propagation direction. Different nonlinear optical crystals have been used for
parametric terahertz wave generation, including LiNbO3 [151], MgO : LiNbO3 [152],
GaSe [153], [154], ZnGeP2 [153], GaP [155], [156], and DAST [157]- [159]. It has
been shown that GaSe is one of the best nonlinear optical crystals, which has the
lowest absorption coefficient in the terahertz domain and has large nonlinear coefficient [153].
Fig. 2.1(a) shows the schematic of a typical terahertz parametric oscillator (TPO)
[151]. The pump laser is a bulky near infrared Q-switched Nd:YAG pulse laser
with a pulse duration of 5-25 ns and a repetition rate of 10-50 Hz. The generated
13
Table 2.4: Characteristics of state-of-the-art terahertz parametric oscillators
Nonlinear optical crystal
peak power
Average power
Frequency
GaSe [161]
GaSe [161]
DAST [158]
DAST [159]
GaP [162]
GaP [156]
GaP [156]
MgO : LiNbO3 [151]
GaP [163]
2364 W
209 W
1-13.4 W
2.8 W
15.6 W
0.66 W
0.62 W
100 mW
100 mW
118.2 µW
10.45 µW
0.2-2.7 µW
0.45 µW
0.78 µW
0.1 µW
94 nW
45 nW
6 nW
51 THz
1.53 THz
2-20 THz
2.5 THz
1.73 THz
1.9 THz
2.6 THz
1.58 THz
2.5-5.6 THz
terahertz signal is a pulse with a duration of 5-10 ns, which can be consider as a
continuous-wave signal in the terahertz frequency range. A TPO is continuously
tunable in the 1-3 THz range with spectral linewidth of tens of GHz. It can emit
peak powers of up to several tenths of a mW at room temperature.
Fig. 2.1(b) shows the schematic of an injection-seeded terahertz parametric generator (IS-TPG) [151], [160]. The pump is a single longitudinal mode Q-switched
Nd:YAG laser working at 1064 nm. A CW Yb-fiber laser (1070 nm) or a tunable
laser (1066-1074 nm) is used for seeding the idler. By introducing injection seeding to the idler, the spectrum linewidth of the generated terahertz signal can be
narrowed to the Fourier transform limlit of the pulse width (less than 200 MHz)
and the output power can be increased to several hundreds higher than that of
TPOs [151]. A terahertz frequency tuning from 0.7 THz to 2.4 THz is possible
using an external cavity laser diode as a tunable seeder.
The characteristics of the state-of-the-art terahertz parametric oscillators are given
in Table 2.4.
14
2.7
Terahertz Photomixers
A terahertz photomixer is an optical heterodyne scheme, in which outputs of two
CW single-mode lasers or output modes of a dual-mode laser with their frequency
difference falling in the terahertz range mix in a nonlinear medium, such as a
photoconductor or a superconductor, and generate a signal, whose frequency is
equal to the frequency difference of the two lasers or two modes of the dual-mode
laser. The generated signal can be coupled either to a transmission line for circuit
applications or to an antenna to radiate the terahertz energy to the free space.
The frequency of the generated terahertz signal can be tuned by tuning the central
frequency of the lasers. Photomixers have the greatest tuning range among all the
coherent sources in terahertz region.
The successfully developed terahertz photomixers to date are the antenna coupled
photomixers, in which two CW laser beams are focused onto an area made of an
appropriate dc biased ultra-fast photoconductor to generate carriers between closely
spaced interdigitated electrodes connected to an antenna. Fig. 2.2 shows a typical
interdigitated-electrode photomixer coupled to a dipole antenna [62]. For wideband
applications bow-tie antenna or self-complementary log-spiral antenna have been
used [42], [45], however, the maximum radiated power from these antennas is less
that that of the resonant antennas. Low-temperature grown (LTG) GaAs has
been widely used as the ultra-fast photoconductor in terahertz photomixers owing
sub-picosecond carrier lifetime, high electric breakdown field (> 50 V/µm), and
relatively high carrier mobility (> 100 cm2 /Vs). ErAs:GaAs also shows promising
characteristics for terahertz photomixer applications [56], [63].
Shown in Fig. 2.3 is the equivalent circuit model for an antenna coupled photomixer
device [65]. The terahertz power coupled to the antenna can be simply modeled
15
Figure 2.2: (a) Top view of interdigitated-electrode vertically-driven photomixer
coupled to a planar dipole antenna (b) Cross-sectional view of photomixer showing
bottom-side coupling of THz radiation through a dielectric lens to free space.
as [39], [65]
PT Hz (ω) =
2RA VB2 mP1 P2 h ηµe i2
1
τ2
2
A
rhν 1 + (ωτ ) 1 + (ωRA C)2
(2.1)
where RA is the antenna’s driving-point resistance, VB is the DC bias, m is the
modulation contrast, P1 and P2 are the power of the two lasers, A is the active
area, η is the external quantum efficiency, µ is the carrier mobility, e is the change
of electron, r is the width of the photoconductive gap, hν is the mean photon energy
of the laser beam, ω is the beat frequency, τ is the carrier recombination lifetime,
and C is the electrode capacitance.
As it can be seen from Eq. 2.1, at high frequencies the terahertz power decreases
τ2
, is inevitable,
with frequency as 1/ω 4 . The carrier lifetime roll-off term,
1 + (ωτ )2
but the other roll-off term due to the RC time constant can be avoided by operating the resonant antenna slightly away from its resonant frequency to cancel out
the capacitive reactance of the photomixer by the antenna inductance [59]. It has
16
Figure 2.3: Equivalent circuit diagram for the photomixer and antenna. Current
drawn from the bias source is modulated at angular frequency ω by the photoconductor, with resistance R(ω, t). The capacitance C represents the effects of the
charge accumulating at the electrodes. Power is dissipated in the antenna, RA (ω),
of which the component oscillating with angular frequency ω is coupled out as
terahertz radiation.
been shown that the RC time constant can also be bypassed in a traveling-wave
photomixer structure [74] (see Fig. 2.4). In this structure, the generated terahertz
signal couples to a transmission line structure if the propagation velocity of the line
is matched to that of the terahertz signal.
The main drawback of the photomixers is their low output power at frequencies
above 1 THz. Typical optical-to-electrical conversion efficiencies are below 10−5
for a single device and output power falls from ≈ 2 µW at 1 THz to below 0.1
µW at 3 THz [59], [63]. The maximum available terahertz power from a single
photomixer device is limited by its maximum sustainable optical power and dc
bias before device failure [78]. The total optical power that a photomixer made
of LTG-GaAs can withstand at room temperature when biased at 30 V is about
0.9 mW/µm2 [46]. The maximum sustainable optical power can be increased by
growing the LTG-GaAs film on a higher-thermal-conductivity substrate such as
17
Figure 2.4: (a) Velocity-match of laser interference fringes to the THz wave on
the coplanar stripline is obtained by a tuning-angle between the two laser beams
(b) Scanning electron microscope picture of a traveling-wave metal-semiconductormetal (TW-MSM) structure with inner part of bow-tie antenna (c) A section of
TW-MSM.
silicon or diamond or by growing a heat spreader epilayer such as AlAs beneath
the LTG-GaAs film [78]. To increase the laser-power-handling capability of the
device, a traveling-wave photomixer has been proposed, in which the optical excitation is distributed over a bigger area [49], [74]. The other method for increasing
the available terahertz power from the photomixers is confining the exciting laser
beams close to the surface of the active layer by backing it up with a dielectric
mirror to form an optical cavity between the top semiconductor/air interface and
the dielectric mirror [73]. One can also increase the available terahertz power by
making an array of photomixer elements [62].
Although most of the proposed terahertz photomixers use lasers with wavelengths
between 750 and 850 nm, however, new materials that enable operation in the optical fiber communications band (1.3-1.55 µm) have been proposed [164]- [169]. The
18
two most promising materials are InGaAs containing ErAs particles [166], [167] and
GaSb containing ErSb nanoparticles [165].
To date, the design optimization of the photomixers has been done based on a
simplified model represented by Eq. (2.1). Although this simple model gives a
good insight on the physics of the photoconductive photomixers and represents
the main aspects of these devices, however, it does not predict all the behaviors
of these devices that have been observed in experiments. In this dissertation, we
develop a more comprehensive model for the photoconductive photomixers to facilitate the device design and optimization processes from material point of view. We
also develop a comprehensive model for the superconductive photomixers, which
are shown that are promising high-power terahertz sources. We also propose the
integrated photomixer-antenna element, for which there is no RC time constant
roll-off term and can be easily used in an array structure to increase the available
terahertz power.
Fig. 2.5 wraps up this chapter by comparing the output power of different continuouswave terahertz sources versus frequency.
19
Gyrotron
4
10
FEL
2
10
QCL (80 K)
0
Power (W )
10
IMPATT
BWO
FIR−Gas Laser
QCL (313 K)
Gunn
−2
10
Gunn
TUNNETT BWO
Photomixer
IMPATT
QCL (48 K)
QCL (78 K)
TPO
−4
10
TUNNETT
Multiplier
TPO
Multiplier
Photomixer
−6
10
TPO
RTD
Photomixer
−8
10
0.1
1
10
100
Frequency (THz)
Figure 2.5: Output power of different continuous-wave terahertz sources versus
frequency.
20
Chapter 3
Photomixing in Superconductors
3.1
Introduction
Exciting a superconductor material with a pair of interfering laser beams, whose
frequency difference is in the terahertz range and have high energy photons (higher
than binding energy of the Cooper pairs), modulates the density of quasiparticles
and Cooper pairs in time and space via Cooper pair breaking effect. In the presence of a dc bias current, the modulated carrier densities generate a modulated
photocurrent inside the superconductor material, which has a frequency harmonic
in the terahertz range in its spectrum. The generated photocurrent can be either
guided down to a transmission line for circuit applications or it can be radiated out
by coupling to an antenna or by proper designing of the superconductor film to act
as an efficient radiator.
21
3.2
Modeling of Superconductive Photomixers
Fig. 3.1 shows a typical photomixing scheme, in which two linearly polarized laser
beams are applied to a dc current biased superconductor film located above a dielectric substrate. The laser beams are considered to be local plane-waves at the
superconductor surface and are assumed to shine the entire surface of the superconductor film. The bath temperature, dc current, and current density are assumed
to be below their critical values for the superconductor material. The associated
electric fields for the two laser beams can be written as
E1(2) = |E1(2) |e
¡
¢
j ω1(2) t−k1(2) ·r
(3.1)
where ω1(2) and k1(2) are the lasers’ angular frequencies and phase constants, respectively, and r is the position vector. The incident optical power is proportional
to the square of the total incident electric field
Pinc (r, t) ∝ |E1 |2 + |E2 |2 + 2|E1 ||E2 | cos(Ωt − K · r)
(3.2)
where Ω = ω1 − ω2 is the angular beat frequency and K = k1 − k2 is the grating
vector. For the coordinate system in Fig. 3.1 the grating vector, K, can be written
as
K = Kx x̂ + Ky ŷ + Kz ẑ
(3.3)
Kx = k2 sin θ2 cos φ2 − k1 sin θ1 cos φ1
(3.4)
Ky = k2 sin θ2 sin φ2 − k1 sin θ1 sin φ1
(3.5)
Kz = k2 cos θ2 − k1 cos θ1
(3.6)
22
z
Laser 1
Laser 2
(ω1 ,k1 )
(ω2 ,k2 )
θ
H
Idc
TS
fil
m
x
(Ω,K)
d
Substrate
Figure 3.1: A typical superconductive photomixer scheme.
where θi and φi , i = 1, 2, are the azimuthal and polar angles of the two beams,
respectively. The beat frequency is assumed to be less than the gap frequency of the
superconductor material to avoid Cooper pair breaking effect due to the generated
signal.
The incident optical power given by Eq.
(3.2) is partially reflected from the
superconductor-air surface and the rest is transmitted into the superconductor film.
For the film thickness in the range of the optical penetration depth, the most part
of the transmitted power is being absorbed by the superconductor material and a
negligible amount of the power reaches to the superconductor-substrate interface.
Hence the optical power inside the superconductor film can be written as
P (r, t) = P0 (1 − R)[1 + η cos(Ωt − K · r)]eαz
(3.7)
where P0 is the total power of the two lasers, R is the optical reflectivity, 0 < η ≤ 1
is the modulation index or grating contrast, and α is the optical absorption coefficient of the superconductor material.
The optical power given by Eq. (3.7) is being absorbed by the electrons via electron-
23
photon interaction and generates high energy quasiparticles by breaking Cooper
pair bounds and also by exciting low energy quasiparticles. The high energy quasiparticles lose their energy via electron-electron and electron-phonon interaction.
The energy does not dissipate in these mechanisms. The transferred energy to the
phonons can be returned back to the electrons by electron-phonon interaction or it
can be dissipated by phonon escape to the substrate. The quasiparticle and phonon
energy spectrum upon optical illumination can be well described by Fermi-Dirac
distribution function [170], which can be modeled by equivalent electron, Te , and
phonon, Tph , temperatures, respectively. The change in the electron and phonon
temperatures can be measured as impedance change in the superconductor film
using modulated reflectivity or photoresponse (PR) methods [171].
The above-mentioned nonequilibrium mechanism can be modeled by a pair of coupled partial differential equations based on the two-temperature model [170]- [173]
¤ P (r, t)
Ce £
d
T̂e (r, t) − T̂ph (r, t) +
T̂e (r, t) =Ke ∇2 T̂e (r, t) −
(3.8)
dt
τe−ph
V Tc
¤ Cph £
¤
d
Ce £
Cph T̂ph (r, t) =Kph ∇2 T̂ph (r, t) +
T̂e (r, t) − T̂ph (r, t) −
T̂ph (r, t) − T̂o
dt
τe−ph
τes
Ce
(3.9)
where T̂e (r, t) and T̂ph (r, t) are the reduced electron and phonon temperatures, Ce
and Cph are the electron and phonon heat capacities, Ke and Kph are the electron
and phonon thermal conductivities, τe−ph is the electron-phonon relaxation time,
τes is the phonon escape time to the substrate, T̂0 is the reduced bath temperature,
Tc is the critical temperature, and V is the volume of the superconductor film. The
terms at the left hand sides of Eqs. (3.8) and (3.9) are the rate of the energy storage
in electron and phonon subsystems, respectively. The first terms at the right hand
sides of Eqs. (3.8) and (3.9) represent the thermal diffusion mechanism and the
24
second terms represent the interaction of electron and phonon subsystems. The
absorbed optical power comes into the equation for the electron subsystem, since
the optical power is being absorbed by the electrons. The last term at the right
hand side of Eq. (3.9) represents the phonon scape to the substrate.
In writing the Eqs. (3.8) and (3.9), we assume that the establishment of the electron temperature occurs during a time smaller than τe−ph and that within this time
almost no energy is transferred from electrons to phonons. Although this assumption may not be valid for YBCO film, however, the two-temperature model provides
a good description of the experimental results on nonequilibrium photoresponse of
high-temperature superconductors [172].
Since the absorbed optical power contains a time-invariant as well as a time-varying
component in its expression, the electron and phonon temperatures will also have
time-invariant and time-varying components in their general expressions. Solving
the coupled differential equations (3.8) and (3.9) with the absorbed optical power
given by Eq. (3.7) results in the reduced electron and phonon temperatures as
e
e
cos(Ωt − K · r − φe )
+ T̂ac
T̂e (r, t) = T̂dc
ph
ph
T̂ph (r, t) = T̂dc
+ T̂ac
cos(Ωt − K · r − φph )
(3.10)
(3.11)
Where
1 ´ αz
P0 (1 − R) ³ 1
+
e
V Tc
Gph Ge−ph
2
2 ´1
+ βph
ηP0 (1 − R) ³ αph
2 αz
=
e
2
2
V Tc
a +b
P0 (1 − R) 1 αz
e
=T̂0 +
V Tc
Gph
ηP0 (1 − R) Ge−ph αz
√
e
=
V Tc
a2 + b 2
e
=T̂0 +
T̂dc
(3.12)
e
T̂ac
(3.13)
ph
T̂dc
ph
T̂ac
25
(3.14)
(3.15)
in which
αph =Kph (|K|2 − α2 ) + Ge−ph + Gph ≈ Ge−ph
αe =Ke (|K|2 − α2 ) + Ge−ph ≈ Ge−ph
βph =ΩCph + 2αKz Kph ≈ ΩCph
βe =ΩCe + 2αKz Ke ≈ ΩCe
a =αe αph − βe βph − G2e−ph ≈ −Ω2 Ce Cph
b =βph αe + βe αph ≈ ΩGe−ph (Ce + Cph )
¡b¢
φph = tan−1
a
³β ´
ph
φe =φph + tan−1
αph
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
where Gph = Cph /τes is the phonon mismatch coefficient and Ge−ph = Ce /τe−ph
is the electron-phonon coupling coefficient. Note that both the electron and the
phonon temperatures decrease along the thickness of the superconductor film, because less optical power absorption takes place at the points farther from the surface
of the superconductor film.
Before applying the optical excitation, the electrons are in equilibrium with the
phonons and with the substrate at the bath temperature, T0 . Upon absorption of
the optical power by the electrons, the electron and phonon temperatures increase
above the bath temperature. The time-invariant component of the absorbed power
causes an equal temperature increase for both the electron and the phonon subsystems. Knowing that Ge−ph ≫ Gph , it can be seen from Eqs. (3.12) and (3.14) that
ph
e
T̂dc
is almost equal to T̂dc
, and both are proportional to the phonon escape time,
τes . This is reasonable, since under the influence of a time-invariant optical power
the electrons and phonons come into equilibrium in a time scale much shorter than
26
the time that the phonons need to escape to the substrate. The shorter the phonon
escape time, the smaller the excess temperature of the electrons and phonons.
The time-varying components of the electron and phonon temperatures are not
equal. The amplitude of the time-varying component of the electron temperature,
e
ph
T̂ac
, is higher than that of the phonon temperature, T̂ac
, because of the higher heat
e
ph
capacity of the phonons. An approximate expression for T̂ac
and T̂ac
from Eqs.
(3.13) and (3.15) can be written as
1
ηP0 (1 − R) 1
αz
1 e
2
V Tc
ΩCph [1 + Ω2 τe−ph
2
]
"
# 21
2
2
+ Ω2 τe−ph
ηP0 (1 − R) 1 Ce2 /Cph
≈
eαz
2
V Tc
ΩCe
1 + Ω2 τe−ph


1


,
Ωτe−ph ≪ 1

Ω


ηP0 (1 − R) αz
≈
e
τe−ph , Ce /Cph < Ωτe−ph < 1

V Tc Cph




 Cph 1 , Ωτe−ph ≫ 1
ph
≈
T̂ac
e
T̂ac
(3.24)
(3.25)
Ce Ω
Comparing Eqs. (3.24) and (3.25), one can show
e
T̂ac
2
i 12
Cph
ph
2 2
≈ 1 + 2 Ω τe−ph T̂ac
Ce
h
(3.26)
From Eq. (3.26), it can be seen that the higher the value of Ωτe−ph the bigger the
e
ph
difference between T̂ac
and T̂ac
. The higher value of Ωτe−ph means that it takes
longer time for the electrons to be in equilibrium with the phonons.
e
The excess time-invariant electron temperature, (T̂dc
− T̂0 ), is much more higher
than the amplitude of the time-varying component of the electron temperature.
27
From Eqs. (3.12) and (3.13) and for η = 1, one can show
"
# 12
2 2
´
1
+
Ω
τ
C
e
e−ph
e
e
T̂ac
T̂dc
− T̂0 ≈ Ωτes
2
2
+ Ω2 τe−ph
Cph Ce2 /Cph
³ Cτ
´
e es
≈
T̂ e ,
Ce /Cph < Ωτe−ph < 1
Cph τe−ph ac
³
(3.27)
The phonon escape time, τes , is in the order of nanoseconds, while the electrone
phonon relaxation time, τe−ph , is in the order of sub-picosecond, hence T̂dc
− T̂0 ≫
e
T̂ac
. This means that the most part of the absorbed optical power is being spent
to increase the time-invariant part of the electron temperature.
Generally, the electron-phonon relaxation time is a function of temperature, especially for the temperatures close to the critical temperature of the superconductor
material [174], [175]. To have a precise solution to the Eqs. (3.8) and (3.9), one
has to include the temperature dependency of the electron-phonon relaxation time
in these equations. The resulting nonlinear equations are difficult to solve. A good
approximation for the electron and phonon temperatures are the solutions given
by Eqs. (3.10) to (3.23), where τe−ph is calculated from the following empirical
formula [175]
τe−ph = τe−ph (Tc )
¤
£ ph 1−γ
ph
1 + β (T̂dc
)
− T̂dc
ph
T̂dc
,
ph
T̂dc
≤1
(3.28)
where τe−ph (Tc ) is the momentum relaxation time at the critical temperature and
ph
β is the residual resistance rate. Note that T̂dc
is independent of τe−ph and is given
by Eq. (3.14).
One can define critical optical power as a power at which the maximum value of
the electron temperature becomes equal to the critical temperature, Tc . Using Eq.
28
(3.10), the critical optical power can be written as
#
"
³ α2 + β 2 ´ 12 −1
V (Tc − T0 ) −αz 1
1
ph
ph
Pc =
e
+
+η
1−R
Gph Ge−ph
a2 + b 2
≈
V (Tc − T0 )
Gph e−αz
1−R
(3.29)
In order to stay in the superconducting region, the total lasers power, P0 , must be
less than the critical optical power, Pc .
Based on the two-fluid model [176], one can write the total photocurrent inside the
superconductor film as
J(r, t) = Jn (r, t) + Js (r, t)
(3.30)
where Jn (r, t) and Js (r, t) are the densities of the normal-current (flow of the quasiparticles) and super-current (flow of the Cooper pairs), respectively. When there
is no optical excitation, the super-current is almost equal to the dc bias current
(Idc ) and the normal-current is too small. At the presence of the optical excitation,
the absorbed optical power breaks some of the Cooper pairs and hence the supercurrent fraction decreases, while the remaining Cooper pairs only participate in dc
current flow mechanism to minimize the ohmic loss. Since the total dc current must
be equal to the dc bias current, Idc , the quasiparticles compensate for the decrease
in the dc current and also carry the generated terahertz photocurrent. The density
of the quasiparticles and the Cooper pairs are assumed to be controlled by the
electron temperature.
The Cooper pairs undergo a collision-less motion under the influence of the applied
bias current, which can be modeled by London’s first equation [176]
E(r, t) = µ0
¤
∂£ 2
λ (r, t)Js (r, t)
∂t
29
(3.31)
where E(r, t) is the electric field inside the superconductor film and λ(r, t) is the
London penetration depth given by [175]
λ2 (r, t) =
λ2L (0)
1 − [T̂e (r, t)]γ
(3.32)
where λL (0) is the London penetration depth at zero temperature and γ = 1.3 to
2.1 is an exponent.
The super-electron density according to the above mentioned model is a dc current
in the direction of the applied dc bias current
Js (r, t) = J0
(3.33)
The normal current density is the result of the drift and diffusion mechanisms of
the quasiparticles
Jn (r, t) = enn (r, t)vn (r, t) + eDn ∇nn (r, t)
(3.34)
where e is the electron charge, nn (r, t) is the quasiparticle density, vn (r, t) is the
velocity of the quasiparticles, and Dn is the quasiparticle diffusion coefficient. The
quasiparticle density in terms of the reduced electron temperature is given by [175]
nn (r, t) = n0 [T̂e (r, t)]γ ,
T̂e (r, t) ≤ 1
(3.35)
where n0 is the total electron density.
The motion of the quasiparticles under the influence of the electric field, E(r,t),
30
inside the superconductor material can be modeled by [176]
E(r, t) =
m ∂vn (r, t)
m
+
vn (r, t)
e
∂t
eτe−ph
(3.36)
where m is the mass of an electron. Note that the electric field and the drift velocity
vectors are assumed to be in the same direction for convenience.
Note that the London’s first equation (Eq. 3.31) has also a nonlinear term in its
general form, which comes from the vs × B term in the Lorentz’s law for the superelectron motion equation [176]. Also Eq. (3.36) needs an vn × B term on its left
hand side coming from Lorentz’s law. These additional terms are perpendicular to
the direction of the applied dc bias current and do not affect the photocurrent element in the direction of the bias current. Hence the equations given by (3.31)-(3.36)
are a complete set of equations that must be solved in order to find the dominant
photocurrent element inside the superconductor film.
e
e
Knowing that T̂ac
≪ T̂dc
and the fact that the average of the total current inside the
superconductor film is equal to the dc bias current (there is no injection of excess
charge into the superconductor film), one can solve Eqs. (3.31) to (3.36) to find
the total photocurrent as
˜ jΩt }
J(r, t) = Jdc + ℜe{Je
(3.37)
where
Idc
wd
¡ 2
¢1
J˜ = J1 + J22 2 e−jφ e−jK·r
Jdc =
J1 =
e
e 2γ−1
(T̂dc
)
µ0 σn τe−ph Ω2 λ2L (0) γ T̂ac
J0
2
2
e
1 + Ω τe−ph
[1 − (T̂dc )γ ]2
31
(3.38)
(3.39)
(3.40)
e
e 2γ−1
µ0 σn Ωλ2L (0) γ T̂ac
(T̂dc
)
J2 = −
J0
2
2
e
1 + Ω τe−ph [1 − (T̂dc )γ ]2
e γ−1 e
+ en0 γDn Kx (T̂dc
) T̂ac
"
#−1
e 2
e 2γ
) (T̂dc
)
µ0 σn τe−ph Ω2 λ2L (0) γ 2 (T̂ac
J0 = Jdc 1 +
2
e γ 2
2(1 + Ω2 τe−ph
) [1 − (T̂dc
) ]
φ = φe + tan−1
J2
J1
(3.41)
(3.42)
(3.43)
in which σn = n0 e2 τe−ph /m is the normal conductivity.
The generated terahertz power inside the superconductor film can be calculated as
PT Hz
1
=
T
Z
0
T
#
" ZZZ
E(r, t)J(r, t) dx dy dz dt
V
(3.44)
where T = 2π/Ω is the temporal period of the generated terahertz signal. One can
calculate the electric field, E(r, t), from Eqs. (3.10) and (3.31) to (3.33) as
E(r, t) = µ0 γΩλ2L (0)
e
e γ−1
T̂ac
(T̂dc
)
e γ 2
[1 − (T̂dc
) ]
sin(Ωt − K · r − φe )
(3.45)
Substituting Eqs. (3.37) and (3.45) into Eq. (3.44) and doing the integration results
in
PT Hz
1
= lwµ0 γΩλ2L (0)
2
0
Z
−d
e
e γ−1
T̂ac
(T̂dc
)
e γ 2
[1 − (T̂dc
) ]
J0 (z)J2 (z) dz
(3.46)
The optical-to-electrical conversion efficiency can be defined as
ηc =
PT Hz
P0
32
(3.47)
3.3
Simulation Results for an YBCO Photomixer
High-temperature superconductors are more desirable for terahertz photomixer applications because of their higher gap frequencies and higher critical temperatures
compare to the low-temperature superconductors. The physical parameters of a superconductive photomixer made of YBa2 Cu3 O7−δ thin film, which is located above
an LaAlO3 substrate with relative permittivity of ǫr = 24 are given in Table 3.1.
The two lasers are located in (x−z) plane and are symmetric with respect to z-axis.
The central wavelength of one of the lasers is fixed at 532 nm. The other laser is
a tunable laser, which its central wavelength can be tuned around 532 nm. The
desired terahertz beat frequency is the result of the frequency difference of the two
lasers. The angle between the two laser beams is θ = 10o and the lasers modulation index is η = 1. At the beat frequency equal to 1 THz, the resulting grating
vector components are Kx = 1.402 × 106 rad/m, Ky = −1.825 × 103 rad/m, and
Kz = −2.086 × 104 rad/m. The dc current bias is 23.8 mA.
Fig. 3.2 shows critical optical power versus distance from the surface of the superconductor film at different bath temperatures. The critical optical power is
minimum at the surface of the film, since most of the optical power is being absorbed at the surface and hence the electron temperature is maximum at this point.
It also can be seen that for the higher bath temperatures the critical optical power
is smaller. The total power of the lasers must be smaller than the critical optical
power at the surface of the superconductor film.
Amplitude of the time-varying components of the electron and phonon temperatures at the surface of the superconductor film (z = 0) are shown in Fig. 3.3 versus
beat frequency. The total laser power is fixed at 150 mW, which is lower than
the minimum of the critical optical power at T0 = 82 K (Fig. 3.2). At low fre-
33
Table 3.1: Physical parameters of a photomixer made of YBa2 Cu3 O7−δ
Description
Notation
Value
Critical temperature [177]
Tc
87 K
Critical current [177]
Ic
25 mA
Critical current density [177]
Jc
9.6 × 105 A/cm2
Exponent [175]
γ
1.68
Residual resistance rate [175]
β
10
London penetration depth [175]
λL (0)
220 nm
Optical penetration depth [177]
δ
90 nm
Optical reflectivity [177]
R
0.1
Optical absorption coefficient [178]
α
1.1 × 105 cm−1
Electron heat capacity [172]
Ce
0.022 Jcm−3 K−1
Phonon heat capacity [177]
Cph
0.85 Jcm−3 K−1
Electron thermal conductivity [179]
Ke
0.4 Wm−1 K−1
Phonon thermal conductivity [179]
Kph
0.1 Wm−1 K−1
Phonon escape time [177]
τes
12 ns
Electron-phonon relaxation time [175] τe−ph (Tc )
3.57 × 10−14 s
Normal state conductivity [175]
σn (Tc )
1.8 × 106 (Ωm)−1
Superconductor film thickness [177]
d
130 nm
Superconductor film length [177]
l
200 µm
Superconductor film width [177]
w
20 µm
Quasiparticle diffusion coefficient [180]
Dn
24 cm2 /s
Total electron density [181]
n0
0.4 × 1022 cm−3
34
3
c
Critical optical power, P (W)
2.5
T
0
=7
K
0
2
7
=7
K
82
T 0=
K
T0
1.5
1
0.5
0
0
20
40
60
80
Depth (nm)
100
120
140
Figure 3.2: Critical optical power versus distance from the surface of the superconductor film at different bath temperatures. The critical temperature is Tc = 87 K.
35
quencies and for smaller electron-phonon relaxation times, the quasiparticles have
enough time to lose their excess energy to the phonons and be in equilibrium with
the phonons. At higher frequencies the quasiparticles can not deliver their excess
energy to the phonons as their temperature changes rapidly with time and hence
they are not in equilibrium with the phonons. The difference between the electron
and phonon temperatures is bigger at higher frequencies and for higher electronphonon relaxation times. The phonon temperature decreases by frequency by 1/Ω
for Ωτe−ph ≪ 1 and can turn to the rate of 1/Ω2 if Ωτe−ph ≫ 1 (see Eq. (3.24)).
The electron temperature decreases by frequency by 1/Ω for Ωτe−ph ≪ 1. As the
beat frequency increases the electron temperature becomes almost independent of
the frequency when Ce /Cph < Ωτe−ph < 1 and then it returns back to the rate of
1/Ω for Ωτe−ph ≫ 1 (see Eq. (3.25)).
Shown in Fig. 3.4 is amplitude of the time-varying component of the electron temperature versus bath temperature at different beat frequencies. For the most values
e
of the beat frequencies and the bath temperatures, Tac
decreases when the bath tem-
perature increases. It can be seen from Eq. (3.25) that for Ce /Cph < Ωτe−ph < 1 the
electron temperature is proportional to the electron-phonon relaxation time, which
decreases when the bath temperature approaches to the superconductor critical
e
temperature. There are two extreme ranges for Ωτe−ph , for which Tac
is indepen-
dent of τe−ph and hence is independent of the bath temperature.
The drift and diffusion current components given at the right hand side of Eq.
(3.41) are depicted versus beat frequency for two different bath temperatures in
Fig. 3.5. As it can be seen from Fig. 3.5, for the electron temperatures far below
the critical temperature, the diffusion current is dominant (solid curves), while for
the electron temperatures close to the critical temperature the drift current becomes
dominant (dashed curves). Note that one can push the electron temperature close
36
to its critical value by either increasing the bath temperature T0 or by increasing
the optical power P0 .
Fig. 3.6 shows total photocurrent at the surface of the superconductor film versus beat frequency for θ = 10o and for different bath temperatures. At the low
temperatures, the diffusion current is dominant and the total current decreases by
frequency. As the bath temperature increases, the drift current starts to dominate
the total current mostly at the higher beat frequencies. At the high enough bath
temperatures, where the electron temperature is close to its critical value, the drift
current dominates the total photocurrent and the current increases with beat frequency. The diffusion current is also a function of the angle between the two laser
beams via Kx . At the limit of θ = 0, where there is no spacial change for the
quasiparticle density, the diffusion current becomes zero. Fig. 3.7 shows the total
photocurrent at the surface of the superconductor film for the case of θ = 0. In
this case the photocurrent is only the result of the quasiparticle drift mechanism
inside the superconductor film. The maximum obtainable frequency is limited by
the gap frequency of the YBCO sample, which can be from 5 THz to 30 THz for
different samples.
Shown in Fig. 3.8 is total photocurrent at the surface of the superconductor film
versus bath temperature for different values of beat frequency. One can see that
the amplitude of the terahertz photocurrent super-linearly increases with the bath
temperature for the temperatures close to the critical temperature.
The amplitude of the photocurrent is not uniform along the thickness of the superconductor film. Figs. 3.9 and 3.10 show variation of the amplitude of the
photocurrent along the thickness of the superconductor film
for different values
of beat frequency and at two different bath temperatures. Comparing Figs. 3.9
and 3.10 shows that at the bath temperature T0 = 82 K the amplitude of the pho-
37
∝ 1/Ω
∝τ
e−ph
∝ 1/Ω
10
Te
ac
1
ac
ac
Te , Tph ( µ K )
∝ 1/Ω
ph
Tac
P = 150 mW
0
T = 77 K ( τ
e−ph
≈ 0.087 ps )
T = 82 K ( τ
e−ph
≈ 0.046 ps )
0
0.1
0
2
∝ 1/Ω
0.1
1
Beat frequency (THz)
5
Figure 3.3: Amplitude of the time-varying components of the electron and phonon
temperatures. The dc current bias is 23.8 mA.
38
80
f=10
70
GHz
P0 = 150 mW
60
Ωτ
f=1
<<1
e−ph
00
G
Hz
40
f=500
e
Tac ( µ K )
50
∝ τe−ph
GHz
decreases by T
0
30
20
10
55
f=1 THz
f=5 THz
60
Ωτe−ph>>1
65
70
75
Bath temperature, T (K)
80
85
0
Figure 3.4: Amplitude of the time-varying component of the electron temperature
versus bath temperature at different beat frequencies.
39
3
Drift and diffusion photocurrent (A / mm2)
10
P = 190 mW
0
θ = 10
o
drift current
diffusion current
2
10
1
T = 77 K
10
0
T0 = 82 K
drift current
0
10
0.1
1
Beat frequency (THz)
5
Figure 3.5: Drift and diffusion components of photocurrent given at the right hand
side of Eq. (3.41) for two different bath temperatures. The critical temperature is
Tc = 87 K.
40
2
Amplitude of THz photocurrent (A / mm )
3
10
P0 = 190 mW
θ = 10
o
drift current
T0
=8
2K
diffusion current
2
T =81 K
0
10
T =77 K
0
T =80 K
0
K
T 0=79
1
10
drift current
diffusion current
0.1
1
Beat frequency (THz)
5
Figure 3.6: Total photocurrent at the surface of the YBCO film versus beat frequency for θ = 10o and at different bath temperatures. The critical temperature is
Tc = 87 K.
41
Amplitude of THz photocurrent (A / mm2)
3
10
P = 190 mW
0
θ=0
T 0=
82 K
2
10
81
T 0=
K
T =80 K
0
1
10
T =77
K
0
0
10
0.1
1
Beat frequency (THz)
5
Figure 3.7: Total photocurrent at the surface of the YBCO film versus beat frequency for θ = 0 and at different bath temperatures. The critical temperature is
Tc = 87 K.
42
P = 190 mW
2
Amplitude of THz photocurrent (A / mm )
0
3
10
o
θ = 10
f = 5 THz
f = 2 THz
f = 1 THz
2
f = 0.5 THz
10
drift
1
10
diffusion current
75
76
77
78
79
80
Bath temperature, T (K)
81
82
0
Figure 3.8: Total photocurrent at the surface of the YBCO film versus bath temperature for θ = 10o and at different beat frequencies.
43
tocurrent decreases rapidly along the thickness of the superconductor film, while at
T0 = 77 K the amplitude of the photocurrent decreases slowly along the thickness.
Concentration of the photocurrent near the surface of the superconductor film will
increase the terahertz radiation power if the film is designed to act as an efficient
radiator.
According to Eq. (3.39), the generated terahertz photocurrent inside the superconductor film is a traveling-wave signal associated with e−jK·r term and also has
a constant phase term e−jφ . Fig. 3.11 shows variation of the phase φ with angle
between the two laser beam at different bath temperatures. At lower temperatures,
the diffusion current, which is proportional to the grating vector component Kx , is
dominant and hence φ strongly depends on θ. As the bath temperature increases,
the drift current dominates and the phase φ becomes independent of θ. Also the
components of the grating vector, K, are functions of θ. In chapter 7 we will show
how one can use this property to steer the radiation beam of an array made of
superconductive photomixers.
Generated terahertz power inside the superconductor film is shown in Fig. 3.12 versus total power of the lasers for different bath temperatures. At the high bath temperature drift current dominates and hence terahertz power increases super-linearly
by temperature. The optical-to-electrical conversion efficiency at T0 = 82 K and
for P0 = 190 mW is %0.003.
44
2
Amplitude of THz photocurrent (A / mm2)
10
P0 = 190 mW
T = 77 K
0
f = 0.5
f=1
o
θ = 10
THz
THz
f=2
f=5
THz
THz
1
10
0
20
40
60
80
Depth (nm)
100
120
140
Figure 3.9: Total photocurrent along the thickness of the YBCO film at T0 = 77K.
45
o
θ = 10
drift current ∝
e (T̂ e )2γ−1
T̂ac
dc
e )2 ]γ
[1−(T̂dc
2
10
f=
f=
f=
TH
z
TH
z
z
THz
0
5
TH
0.5
1
10
2
1
f=
Amplitude of THz photocurrent, J
ac
2
(A / mm )
P0 = 190 mW
T0 = 82 K
3
10
20
40
60
80
Depth (nm)
100
120
140
Figure 3.10: Total photocurrent along the thickness of the YBCO film at T0 = 82K.
46
100
f= 1 THz
80
=7
0
T
20
9
0
40
K
=7
7K
60
T
Photocurrent phase, φ ( Deg )
P0 = 190 mW
0K
=8
0
T0
−20
−40
T =81 K
0
−60
0
2
4
6
8
Angle between two lasers, θ ( Deg )
10
Figure 3.11: Phase of photocurrent versus angle between the two laser beams.
47
10
1
T0=82 K
Terahertz Power, P
THz
(µW)
f= 3 THz
0.1
T0=77 K
0.01
150
155
160
165
170
175
Optical Power, P (mW)
180
185
190
0
Figure 3.12: Generated terahertz power inside the YBCO film versus total incident
optical power at two different bath temperatures. The applied dc bias is 23.8 mA
48
Chapter 4
Photomixing in Photoconductors
4.1
Introduction
In a photoconductive continuous-wave terahertz photomixer, the incident optical
power density from two interfering laser beams excites the electrons from valence
band to conduction band and generates spatiotemporal electron and hole concentration gratings. In the presence of a dc voltage bias, the electron and hole densities
generate a current that contains a beat frequency harmonic in its spectrum. One
can solve the continuity equations for the electron and hole densities along with the
appropriate boundary conditions to find the resulting photocurrent.
Low-temperature grown (LTG) GaAs has been predominantly used in terahertz
photoconductive photomixers. LTG-GaAs was discovered at Lincoln Lab by A.
R. Cawala in the mid 1980’s and has been used in variety of electronic and optoelectronic devices and systems since then. LTG GaAs is deposited by molecular
beam epitaxy (MBE) at low substrate temperature ∼ 200 o C in an As-rich environment and then is annealed under an As overpressure at higher temperature
49
∼ 600 o C for about 10 minutes. A typical run includes a growth rate of 1 µm/h
on a (001)-oriented GaAs substrate with an As4 /Ga beam equivalent pressure ratio
of 10 [182]. In LTG-GaAs growing process, because of the low substrate temperature and the As-rich growth conditions, about 1% excess As is incorporated
into the LTG-GaAs lattice. The excess As atoms result in formation of different types of point defects such as As antisites (AsGa : Ga atom located on an As
site in the crystal lattice, or As atom located on a Ga site), As interstitials, and
Ga-related vacancies (VGa ) [182]. Unannealed LTG-GaAs is relatively conducting
material (ρ ∼ 10 Ω cm), while annealed LTG-GaAs is a semi-insulating material
(ρ ∼ 107 Ω cm); a property which along with its relatively high carrier mobility
(µn ∼ 200 cm2 /Vs) and sub-picosecond carrier lifetime characteristics make it an
ideal material for high speed photoconductive switching applications.
When LTG-GaAs is annealed, the concentration of point defects reduces and As
precipitates in the form of As clusters appear [183]. The large excess As concentration in LTG-GaAs results in a large trap density, that consequently results in a high
resistivity, large breakdown field (∼ 50 V/µm), and extremely short carrier lifetime
(∼ 0.2 ps). The carrier lifetime of the LTG-GaAs can be controlled by the annealing and growth conditions [184]- [189]. Because of its unique properties, LTG-GaAs
has been used as terahertz sources and detectors, ultra-fast and high-voltage photoconductive switches, buffer layer that eliminates sidegating and backgating in
GaAs integrated circuits, and high-responsivity, wide-bandwidth photodetectors.
In this chapter, a dc biased photoconducting film excited by two CW laser beams
is considered as a typical photomixing scheme. The analytical expression for the
photocurrent in all directions inside the photoconductor is derived, and its dependency on the applied electric field, material and laser parameters, and photomixer
configuration is explored. The saturation behavior of the photocurrent along with
50
its super-linear variation with the applied electric field is studied in detail. It is
shown that the amplitude and the phase of the photocurrent change with the angle
between the two laser beams, which can be used to develop a photomixer based
phased array antenna.
4.2
Modeling of Photoconductive Photomixers
Fig. 4.1 shows a typical heterodyne photomixing scheme, in which two linearly
polarized laser beams with a difference in their central frequencies falling in the
terahertz spectrum are applied to a dc biased photoconductor film.
The optical power density inside the photoconductor film can be written as
I(r, t) = I0 T (θ)[1 + η cos(Ωt − K · r)]eαz
(4.1)
where I0 is the total optical power density of the two lasers, T (θ) is the optical
power density transmission coefficient at the photoconductor-air interface, θ is the
angle between the two incident laser beams, α is the absorption coefficient, and
0 < η ≤ 1 is the modulation index or grating contrast. As it can be seen from
Eq. (4.1), the absorbed optical power density has a spatiotemporal distribution
inside the photoconductor film, and hence generates a spatiotemporal electron-hole
concentration inside the film.
The continuity equations for the electron and hole densities, N (r, t) and P (r, t),
are [190]
∂N (r, t)
1
= ∇ · Jn (r, t) + G(r, t) − Rn (r, t)
∂t
e
1
∂P (r, t)
= − ∇ · Jp (r, t) + G(r, t) − Rp (r, t)
∂t
e
51
(4.2)
(4.3)
z
Laser 2
Laser 1
(ω1 ,k1 )
(ω2 ,k2 )
θ
x
+V
ǫr (Ω,K)
d
Figure 4.1: A typical photoconductive photomixer scheme.
where e is the electron charge, G(r, t) is the carrier generation rate, Jn (r, t) and
Jp (r, t) are the electron and hole current densities, and Rn (r, t) and Rp (r, t) are
the electron and hole recombination rates, respectively. The generation rate can be
written in terms of the absorbed intensity I(r, t) as
G(r, t) =
α
I(r, t) = Go [1 + η cos(Ωt − K · r)]eαz
ℏω
(4.4)
where ℏ is the reduced Planck constant, ω is the average of the angular frequencies
of the two laser beams, and G0 = αT I0 /ℏω. For the laser excitation with the
intensity around 1 mW/µm2 , the dominant recombination mechanism is the trap
assisted recombination process [191]. Therefore, the recombination rates can be
approximated by [192]
1
N (r, t)
τn
1
Rp (r, t) = P (r, t)
τp
Rn (r, t) =
52
(4.5)
(4.6)
where τn and τp are the electron and hole recombination lifetimes, respectively.
The carrier current densities Jn (r, t) and Jp (r, t) are the sum of the drift and diffusion currents
Jn (r, t) = eN (r, t)vn (r, t) + eDn ∇N (r, t)
(4.7)
Jp (r, t) = eP (r, t)vp (r, t) − eDp ∇P (r, t)
(4.8)
where vn and vp are the electron and hole drift velocities, and Dn and Dp are the
electron and hole diffusion coefficients, respectively.
Generally, the carrier drift velocity and the carrier lifetime are functions of applied
electric field [193]- [197]. Although our analysis is general, the simulation results
are concentrated on LTG-GaAs based photomixers. An analytic expression for the
carrier drift velocity in GaAs in terms of the electric field can be written as [198]
µn0 + vsn (E03 /Ec4 )
E(r, t) = µn (E0 )E(r, t)
1 + (E0 /Ec )4
µp0 + vsp (E03 /Ec4 )
E(r, t) = µp (E0 )E(r, t)
vp =
1 + (E0 /Ec )4
vn =
(4.9)
(4.10)
where E(r, t) = E0 + E(r, t) is the total electric field inside the photoconductor,
Ec = 1.7 V/µm is a fitting parameter to fit Eq. (4.9) to the experimental results
given by Sun et al. for carrier velocity in LTG-GaAs [199], vsn and vsp are the
electron and hole saturation velocities, µn0 and µp0 are the low field electron and
hole mobilities, and µn (E0 ) and µp (E0 ) are the field dependent electron and hole
mobilities, respectively. In Eqs. (4.9) and (4.10) we assume that the amplitude of
the induced electric field, E(r, t), is much smaller than the amplitude of the applied
electric field E0 . This is a reasonable assumption, since the induced electric field
is the result of the electron and hole local separation mechanism, which is a weak
53
phenomena. The applied electric field is a constant vector along the direction of the
applied dc bias, and its amplitude can be calculated by dividing the amplitude of
the applied voltage to the length of the photoconducting film. We also use µn (E0 )
and µp (E0 ) from Eqs. (4.9) and (4.10) to calculate carrier diffusion coefficients, Dn
and Dp , from Einstein relationship [190].
The expression for the carrier lifetime in LTG-GaAs in terms of the applied electric
field can be written as [200]
h
i3
τ (E0 ) = τ (E0 = 0) × (Te /T )1.5 × r(E0 = 0)/r(E0 )
(4.11)
where T is the lattice temperature, Te is the electron temperature, and r is the
radius of the well in the direction of the applied field.
The total electric field, E(r, t), is related to the electron and hole densities by
Poisson’s equation
e
∇ · E(r, t) = [P (r, t) − N (r, t)]
ǫ
(4.12)
where ǫ is the permittivity of the photoconductor.
From Eqs. (4.2) to (4.12), one can extract a set of nonlinear coupled partial differential equations given by Eqs. (4.13) and (4.14). Two nonlinear terms are the first
and the second terms in the right hand sides of the two equations.
∂N (r, t)
e
= µn (E0 )E(r, t) · ∇N (r, t) + µn (E0 ) N (r, t)[P (r, t) − N (r, t)]
∂t
ǫ
1
(4.13)
+ Dn ∇2 N (r, t) − N (r, t) + G(r, t)
τn
e
∂P (r, t)
= −µp (E0 )E(r, t) · ∇P (r, t) − µp (E0 ) P (r, t)[P (r, t) − N (r, t)]
∂t
ǫ
1
+ Dp ∇2 P (r, t) − P (r, t) + G(r, t)
(4.14)
τp
54
One can linearize Eqs. (4.13) and (4.14) by replacing the total electric field, E(r, t),
with the applied electric field E0 and also replacing the electron and hole densities,
N (r, t) and P (r, t), in the first part of the second terms in the right hand sides of
the equations (4.13) and (4.14) with their average values over the thickness of the
photoconducting film.
Note that we do not replace the total electric field by the applied electric field from
the beginning. We have taken into account the effect of the local electron and hole
separation mechanism by incorporating Eq. (4.12).
Applying the above mentioned simplifying assumptions to Eqs. (4.13) and (4.14)
results in a set of linear coupled partial differential equations for the electron and
hole densities as
N
e
∂N
= µn E0 · ∇N + µn N̄0 [P − N ] + Dn ∇2 N −
+G
∂t
ǫ
τn
P
e
∂P
= −µp E0 · ∇P − µp P̄0 [P − N ] + Dp ∇2 P − + G
∂t
ǫ
τp
(4.15)
(4.16)
where
1
N̄0 [P¯0 ] =
(1 − e−αd ) N0 [P0 ]
αd
(4.17)
in which N0 and P0 are the solutions to Eqs. (4.15) and (4.16) associated with the
time independent term of the generation rate function given by Eq. (4.4).
Equations (4.15) and (4.16) are two coupled non-homogeneous parabolic partial differential equations, which are difficult to solve for a complete set of solutions. Since
we are interested in the solution at the steady state, we use the non-homogeneous
ordinary differential equation method to find the solution to these equations [201].
One can rewrite Eq. (4.4) as
G(r, t) = Go eαz + δG(r, t)eαz
55
(4.18)
where
δG(r, t) = ℜe{ηGo ej(Ωt−K·r) }
(4.19)
Hence the particular solution for Eqs. (4.15) and (4.16) can be written as
N p (r, t) = N0 eαz + δN (r, t)eαz
(4.20)
P p (r, t) = P0 eαz + δP (r, t)eαz
(4.21)
ˆ ej(Ωt−K·r) }
δN (r, t) = ℜe{δN
(4.22)
ˆ ej(Ωt−K·r) }
δP (r, t) = ℜe{δP
(4.23)
where
Substituting Eqs. (4.18) to (4.23) into Eqs. (4.15) and (4.16) results in the following
equations for N0 and P0
N0
e
+ G0 = 0
µn N̄0 [P0 − N0 ] + α2 Dn N0 −
ǫ
τn
P0
e
+ G0 = 0
−µp P̄0 [P0 − N0 ] + α2 Dp P0 −
ǫ
τp
(4.24)
(4.25)
A solution for Eqs. (4.24) and (4.25) is given in Appendix A.
ˆ and δP
ˆ are
The resulting equations for δN
i
e
ˆ
Dn (|K|2 − α2 ) + µn N̄0 + τn−1 + jµn (E0 · K) + 2jαDn Kz + jΩ δN
ǫ
i
he
ˆ = ηG0
(4.26)
− µn N̄0 δP
ǫ
i
h
e
ˆ
Dp (|K|2 − α2 ) + µp P¯0 + τp−1 − jµp (E0 · K) + 2jαDp Kz + jΩ δP
ǫ
i
he
ˆ = ηG0
¯
(4.27)
− µp P0 δN
ǫ
h
56
ˆ and δP
ˆ as
One can solve Eqs. (4.26) and (4.27) to find δN
τn
ˆ = |δN
ˆ |ejφn = an + jbn ηG0 ≈
δN
ηG0
a + jb
1 + jΩτn
τp
ˆ = |δP
ˆ |ejφp = ap + jbp ηG0 ≈
ηG0
δP
a + jb
1 + jΩτp
(4.28)
(4.29)
where
e
an =τp−1 + Dp (|K|2 − α2 ) + (µn N̄0 + µp P¯0 ) ≈
ǫ
e
ap =τn−1 + Dn (|K|2 − α2 ) + (µn N̄0 + µp P¯0 ) ≈
ǫ
1
τn
1
τp
(4.30)
(4.31)
bn =Ω − µp (E0 · K) + 2αDp Kz ≈ Ω
(4.32)
bp =Ω + µn (E0 · K) + 2αDn Kz ≈ Ω
¤
e£
1
a =an ap − bn bp − µp an P¯0 + µn ap N̄0 ≈
− Ω2
ǫ
τn τp
¤
e£
1
1
b =ap bn + an bp − µp bn P¯0 + µn bp N̄0 ≈ Ω( + )
ǫ
τn τp
(4.33)
(4.34)
(4.35)
The homogeneous solution to Eqs. (4.15) and (4.16) at the steady state is the
solution to the following coupled homogenous partial differential equations
e
Nh
µn E0 · ∇N h + µn N̄0 [P h − N h ] + Dn ∇2 N h −
=0
ǫ
τn
Ph
e
=0
−µp E0 · ∇P h − µp P̄0 [P h − N h ] + Dp ∇2 P h −
ǫ
τp
(4.36)
(4.37)
Eqs. (4.36) and (4.37) can be combined into a single partial differential equation
as
£
¤ £
¤ ¡ µp µn ¢ h h
N P =0
+
µn µp E0 · P h ∇N h −N h ∇P h + µp Dn P h ∇2 N h +µn Dp N h ∇2 P h −
τn τp
(4.38)
57
One can consider a solution set for Eq. (4.38) as
−1 z
N h (z) = N1 eL
−1 z
P h (z) = P1 eL
(4.39)
(4.40)
where N1 and P1 are two constants that can be determined by applying the appropriate surface recombination boundary conditions. The parameter L in Eqs. (4.39)
and (4.40) is the effective diffusion length that can be found by substituting Eqs.
(4.39) and (4.40) into Eq. (4.38)
L=
√
Dτ
µ p Dn + µ n Dp
µn + µp
µn + µp
τ = µp µ n
+ τp
τn
D=
(4.41)
(4.42)
(4.43)
Knowing the particular and the general solutions to Eqs. (4.15) and (4.16), one
can write the complete solution as
£
¤
−1
N (r, t) = N p (r, t) + N h (z) = N0 + δN (r, t) eαz + N1 eL z
£
¤
−1
P (r, t) = P p (r, t) + P h (z) = P0 + δP (r, t) eαz + P1 eL z
(4.44)
(4.45)
The surface recombination boundary condition can be written as [190], [191]
¯
∂N ¯¯
¯
= −sn N ¯
¯
∂z z=0
z=0
¯
∂P ¯¯
¯
Dp
= −sp P ¯
¯
∂z z=0
z=0
Dn
58
(4.46)
(4.47)
where sn and sp are the electron and hole surface recombination velocities, respectively. Applying Eqs. (4.46) and (4.47) to Eqs. (4.44) and (4.45) yields
sn + αDn
N0
sn + L−1 Dn
sp + αDp
P0
P1 = −
sp + L−1 Dp
(4.48)
N1 = −
(4.49)
The total electron and hole densities can be written as
h
sn + αDn L−1 z i
ˆ | cos(Ωt − K · r + φn )eαz
N (r, t) =N0 eαz −
e
+ |δN
sn + L−1 Dn
h
sp + αDp L−1 z i
ˆ | cos(Ωt − K · r + φp )eαz
e
+ |δP
P (r, t) =P0 eαz −
sp + L−1 Dp
(4.50)
(4.51)
Using Eqs. (4.7) to (4.10) along with the electron and hole densities given by
Eqs. (4.50) and (4.51) results in the expression for the photocurrent inside the
photoconductor as
Jx (r, t) =Jx0 + Jx1 cos(Ωt − K · r) + Jx2 sin(Ωt − K · r)
(4.52)
Jy (r, t) =Jy1 cos(Ωt − K · r) + Jy2 sin(Ωt − K · r)
(4.53)
Jz (r, t) =Jz0 + Jz1 cos(Ωt − K · r) + Jz2 sin(Ωt − K · r)
(4.54)
where Jx , Jy , and Jz are the photocurrent components in the x, y, and z directions,
respectively, and
−1
Jx0 =e(vn N0 + vp P0 )eαz + e(vn N1 + vp P1 )eL z
h
ˆ | cos φn + vp |δP
ˆ | cos φp + Dn Kx |δN
ˆ | sin φn
Jx1 =e vn |δN
i
ˆ | sin φp eαz
− Dp Kx |δP
59
(4.55)
(4.56)
Jx2
Jy1
Jy2
h
ˆ | sin φn − vp |δP
ˆ | sin φp + Dn Kx |δN
ˆ | cos φn
=e − vn |δN
i
ˆ | cos φp eαz
− Dp Kx |δP
i
h
ˆ | sin φn − Dp |δP
ˆ | sin φp eαz
=eKy Dn |δN
h
i
ˆ
ˆ
=eKy Dn |δN | cos φn − Dp |δP | cos φp eαz
−1
Jz0 =eα(Dn N0 − Dp P0 )eαz + eL−1 (Dn N1 − Dp P1 )eL z
h
ˆ | cos φn − αDp |δP
ˆ | cos φp + Kz Dn |δN
ˆ | sin φn
Jz1 =e αDn |δN
i
ˆ | sin φp eαz
− Kz Dp |δP
h
ˆ | sin φn + αDp |δP
ˆ | sin φp + Kz Dn |δN
ˆ | cos φn
Jz2 =e − αDn |δN
i
ˆ
− Kz Dp |δP | cos φp eαz
(4.57)
(4.58)
(4.59)
(4.60)
(4.61)
(4.62)
where vn = µn E0 and vp = µp E0 are the electron and hole drift velocities, respectively, and E0 = E0 x̂. As it can be seen from Eqs. (4.55) to (4.62), the
photocurrent components in the lateral and transversal directions are only diffusion currents, while the longitudinal photocurrent contains both diffusion and drift
components. Hence, the longitudinal component is much greater than the two other
components.
For the case where the two laser beams are normally incident the photoconductor
surface and for τn ≈ τp , the longitudinal component of the photocurrent given by
Eq. (4.52) can be approximately written as
"
³
e
τn v n
Jx (z, t) ≈ T αI0
(τn vn + τp vp ) +
ℏω
[1 + (Ωτn )2 ]1/2
#
´
τp v p
η cos(Ωt − Kz z − φ) eαz
+
2
1/2
[1 + (Ωτp ) ]
60
(4.63)
which has the same form as the photocurrent expression given by E. R. Brown [62].
The equivalent surface current density can be calculated by integrating the volume
current density over the thickness of the photoconducting film. The longitudinal
component of the surface current density is a traveling-wave current given by
s
Js (ρ, t) = Jdc
+ ℜe{J˜s ejΩt }
(4.64)
where
£
¤
s
Jdc
=ed vn (N̄0 + N̄1 ) + vp (P¯0 + P¯1 )
£ 2
¤
2 1/2 −jφ −jK·ρ
J˜s = Js1
+ Js2
e e
h
i
e
n
n
ˆ | + (αp vp + Dp αp Kx )|δP
ˆ|
(α
v
−
D
α
K
)|
δN
Js1 = 2
n
n
x
1
2
1
2
α + Kz2
i
h
e
p
p
n
n
ˆ
ˆ
(α2 vn + Dn α1 Kx )|δN | + (α2 vp − Dp α1 Kx )|δP |
Js2 = 2
α + Kz2
Js2
φ = tan−1
Js1
(4.65)
(4.66)
(4.67)
(4.68)
(4.69)
in which
N̄1 [P¯1 ] =
1
L−1 d
−1 d
(1 − e−L
) N1 [P1 ]
(4.70)
n[p]
= α1 cos φn[p] + α2 sin φn[p]
(4.71)
n[p]
= α2 cos φn[p] − α1 sin φn[p]
(4.72)
α1
α2
α1 = α + Kz e−αd sin Kz d − αe−αd cos Kz d
(4.73)
α2 = −Kz + Kz e−αd cos Kz d + αe−αd sin Kz d
(4.74)
In Eq. (4.66), ρ is the position vector in the x − y plane.
The amplitude of the terahertz photocurrent given by Eq. (4.64) is related to the
61
photoconductor’s physical parameters as well as to the photomixer’s structure by
Eqs. (4.28) to (4.35) and Eqs. (4.67) to (4.74). It can be seen that the amplitude
of the terahertz photocurrent is not only related to the carrier lifetime and the beat
frequency, but also is a function of the other parameters of the photoconductor
material and the photomixer structure.
In the next section, the effects of the different physical parameters as well as the
configuration of the photomixer on the generated photocurrent are studied. In this
analysis we neglect the electric field screening effect inside the photoconductor film.
4.3
Simulation Results for a Photomixer made of
LTG-GaAs
The physical parameters of a typical photomixer made of LTG-GaAs are given in
Table 4.1. The total photocurrent given by Eq. (4.64) contains two components;
s
the dc component, Jdc
, and the terahertz component, J˜s . The dc component of
the photocurrent, which is given by Eq. (4.65), is proportional to the carrier drift
velocities, vn and vp , and the carrier lifetimes, τn and τp , and has no contribution in
terahertz signal generation process. The dc component of the photocurrent limits
the maximum sustainable dc bias and optical power before device failure, and hence
limits the available terahertz signal.
The amplitude of the terahertz photocurrent, which is given by Eqs. (4.66) to
(4.68), is proportional to the carrier drift velocities and the carrier densities, and
also is a function of the geometry of the photomixer and the other physical parameters of the photoconductor. Note that the terahertz photocurrent is related to the
applied electric field via carrier drift velocities and carrier lifetimes.
62
Table 4.1: Physical parameters of a photomixer made of LTG-GaAs
Description
Notation
Value
Work temperature
T0
300 K
Absorption coefficient at λ = 780 nm [190]
α
10000 cm−1
Photoconductor thickness
d
2 µm
Photoconductor length
l
10 µm
Angle between the two laser beams
θ
10◦
Laser central wavelength
λ
780 nm
Laser intensity
I0
0.4 mW/µm2
Electron saturation velocity [73]
vsn
4 × 104 m/s
Hole saturation velocity [73]
vsp
1 × 104 m/s
Low field electron lifetime [73]
τn0
0.1 ps
Low field hole lifetime [73]
τp0
0.4 ps
Low field electron mobility [41]
µn0
400 cm2 /Vs
Low field hole mobility [41]
µp0
100 cm2 /Vs
Surface recombination velocity [202]
sn,p
5000 m/s
Relative permittivity [44]
ǫr
12.8
Fig.
4.2 shows variation of the electron lifetime with applied electric field in
LTG-GaAs material. The graph is generated using the method given in [200].
The effective radius of the ionized donor’s Colombic potential, r, is found as
r(E0 = 0) = 1.1632 nm. As it is expected, the electron lifetime is almost independent of the applied electric field for the low field regime, while it increases
exponentially with the rate of τn ∝ E0 1.8 for high values of the applied electric field.
ˆ |, versus electron lifetime
Shown in Fig. 4.3 is amplitude of the electron density, |δN
for different beat frequencies. It can be seen that the electron density is proportional to the electron lifetime for small values of τn and becomes independent of τn
when τn increases above a certain value. The linear region is bigger for the smaller
values of the beat frequencies and becomes smaller at the higher beat frequencies.
The electron density is related to the applied electric field via carrier lifetime. Since
the electron density is independent of the carrier lifetime for high values of τn (see
63
10
∝
Electron lifetime, τn (ps)
0
E 1.8
τn0 = 0.1 ps
1
1
10
Applied electric field, E (V/µm)
50
0
Figure 4.2: Electron lifetime versus applied electric field in LTG-GaAs. The lowfield electron lifetime is τn0 = 0.1 ps.
64
1.5
E0 = 0.5 V/µm
| δ^ N| (× 1021 m−3)
ˆ ≈
δN
f = 0.1 THz
τn
ηG0
1+jΩτn
1
f = 0.2 THz
0.5
f = 0.5 THz
f = 1 THz
0
0.5
1
1.5
2
2.5
3
Electron lifetime, τn (ps)
Figure 4.3: Amplitude of electron density inside the LTG-GaAs film versus electron
lifetime at different beat frequencies. The total applied optical power density is
I0 = 0.4 mW/µm2 .
65
Fig. 4.3), it is also independent of the applied electric field for high values of E0
as shown in Fig. 4.4. It may worth to note that there is a small range for the
applied electric field before the saturation region, for which the amplitude of the
electron density increases super-linearly with E0 , specially at low frequencies. For
sufficiently high values of the applied electric field, the amplitude of the electron
density is no longer proportional to the carrier life time (see Fig. 4.3) and becomes
independent of it, where the saturation occurs.
Fig. 4.5 shows the dependency of the amplitude of the electron density on the
frequency and electron lifetime. A small value has been chosen for the applied
electric field to keep the low field electron lifetime unchanged. It can be seen from
Fig. 4.5 that the amplitude of the electron density remains between the two upper
and lower bounds depicted by dotted and dashed curves, respectively.
For small
values of τn0 the electron density approaches to its lower bound, which represents
τn
functionality with the electron life time and the beat frequency,
[1 + (Ωτn )2 ]1/2
while for bigger values of τn0 it approaches to its upper bound, which is G0 /Ω. At
high frequencies, the two upper and lower bounds merge together and the electron
density becomes independent of the electron lifetime. The same scenario happens
for the hole parameters.
Amplitude of the terahertz photocurrent in a typical photomixer made of LTGGaAs is shown in Fig. 4.6. As it can be seen from Fig. 4.6, amplitude of the
terahertz photocurrent saturates when the applied electric field increases above a
certain value. This photocurrent saturation behavior is the direct result of the carrier drift velocity saturation phenomena in photoconductors. Recently, the form of
behavior shown in Fig. 4.6 has been experimentally confirmed by Michael et al. for
the terahertz power generated by an LTG-GaAs based photomixer [74].
According to Eqs. (4.67) and (4.68) the terahertz photocurrent can be simply mod-
66
f = 0.1 THz
τn0 = 0.1 ps
τp0 = 0.4 ps
−3
m )
0
∝ E 1.4
1
| δ N| (× 10
21
0.2 THz
^
f = 0.5 THz
f = 1 THz
0.1
o.01
0.1
1
10
Applied electric field, E0 (V/µm)
Figure 4.4: Amplitude of electron density inside the LTG-GaAs film versus applied
electric field at different beat frequencies. The total applied optical power density
is I0 = 0.4 mW/µm2 .
67
E0 = 0.5 V/µm
1
G
Ω 0
τn
G
[1+(Ωτn )2 ]1/2 0
| δ^ N| (× 1021 m−3)
1
τn0 = 0.5 ps
τn0 = 0.25 ps
0.1
τn0 = 0.15 ps
0.5
1
1.5
2
Beat frequency (THz)
2.5
Figure 4.5: Amplitude of electron density versus beat frequency for different electron lifetimes in an LTG-GaAs photomixer. The total applied optical power density
is I0 = 0.4 mW/µm2 .
68
10
v saturates
ˆ |vn
|J˜s | ∝ e|δN
saturates with τ
n
∝
Hz
T
0.1
1.3
E0
0.5 THz
in
c
re
as
es
n
n
v
Amplitude of THz photocurrent (A/m)
ses
crea
τ n in
1
τ
=0
n0
.5
f=
f= 1 THz
ps
1
0.
z
TH
0.1
0.01
τn0=0.1 ps
0.1
1
10
Applied electric field, E (V/µm)
0
Figure 4.6: Amplitude of terahertz equivalent surface photocurrent in LTG-GaAs
photomixer versus applied electric field at different beat frequencies and for two
different low-field electron lifetimes. The total applied optical power density is
I0 = 0.4 mW/µm2 .
69
ˆ |vn . The electron drift velocity, vn , approaches to its saturation
elled as |J˜s | ∝ e|δN
value vsn for high values of the applied electric field. The amplitude of the electron
ˆ | also saturates for high values of the applied electric field, hence the
density, |δN
photocurrent saturation occurs at high values of E0 . Note that the photocurrent
saturation occurs at lower values of the applied electric field for higher frequencies.
One may notice from Fig. 4.6 that the terahertz photocurrent shows a super-linear
variation with the applied electric field before the saturation region; the behavior
that has been experimentally shown by some authors [41], [78]. As it can be seen
from Fig. 4.6, one can achieve a higher level of terahertz photocurrent by increasing the low field carrier lifetime, although increasing the carrier lifetime above 1
ps will not increase the terahertz photocurrent anymore. The carrier lifetime in
an LTG-GaAs sample can be controlled by controlling its growth and annealing
conditions [186].
The dc component of the generated photocurrent is one of the limiting factors of
the achievable terahertz power in a terahertz photomixer [78]. Unlike the terahertz
photocurrent, the dc photocurrent does not saturate with the applied electric field
and also increases by increasing the carrier lifetime. Fig. 4.7 shows the amplitude
of the dc photocurrent versus applied electric field for different electron lifetimes.
As it can be seen from Figs. 4.6 and 4.7, there is a trade-off between the maximum
sustainable dc photocurrent before the device failure and the maximum achievable
terahertz signal in terms of the carrier lifetime and the applied electric field. The
optimum operation condition for an LTG-GaAs photomixer is a dc bias just before
the saturation region and a carrier life time around 0.5 ps.
Fig. 4.8 shows amplitude of the terahertz photocurrent versus beat frequency for
different values of the electron lifetime. Comparing Fig. 4.5 with Fig. 4.8 shows
that the terahertz photocurrent and the carrier density have the same form of
70
1000
Amplitude of DC photocurrent (A/m)
τn0 = 1 ps
τn0 = 0.5 ps
100
τn0 = 0.25 ps
τn0 = 0.1 ps
10
1
0.1
0.01
0.1
1
10
100
Applied electric field, E (V/µm)
0
Figure 4.7: Amplitude of dc photocurrent in LTG-GaAs photomixer versus applied
electric field at different low-field electron lifetimes. The total applied optical power
density is I0 = 0.4 mW/µm2 .
71
Amplitude of THz photocurrent (A/m)
E0 = 0.5 V/µm
τn0 = 1 ps
τn0 = 0.5 ps
τn0 = 0.25 ps
1
τn0 = 0.1 ps
∝ 1/
Ω
0.1
0
0.5
1
1.5
2
2.5
3
Beat frequency (THz)
Figure 4.8: Amplitude of terahertz equivalent surface photocurrent in LTG-GaAs
photomixer versus beat frequency at different low-field electron lifetimes. The total
applied optical power density is I0 = 0.4 mW/µm2 .
72
dependency on the carrier lifetime. The amplitude of the terahertz photocurrent
decreases by the rate of 1/Ω with the beat frequency.
Fig. 4.9 shows amplitude of the terahertz photocurrent versus thickness of the photoconducting film. The amplitude of the terahertz photocurrent increases exponentially for the thicknesses smaller than the penetration depth, 1/α, since the thicker
the photoconductor film the more the optical power absorption. The amplitude
of the terahertz photocurrent saturates for the thicknesses beyond 2/α. Although
Fig. 4.9 suggests higher thicknesses for the photoconducting film to achieve higher
terahertz power, however the thicker photoconductor has smaller thermal conductivity [78], which can limit the maximum applicable optical power.
Fig. 4.10 shows variation of the longitudinal component of the grating vector, Kx ,
with angle between the two lasers. By changing Kx , one can control the phase
distribution of the traveling-wave terahertz photocurrent (see Fig. 4.10), which
can be used to design an array of the photomixer elements with beam steering
capability [83].
73
1
Amplitude of THz photocurrent (A/m)
Beat frequency = 1 THz
0.1
α = 20000 cm−1
α = 14000 cm−1
α = 10000 cm−1
0.01
α = 6000 cm−1
0
0.5
1
1.5
2
2.5
3
3.5
4
Thickness of photoconducting film, d (µm)
Figure 4.9: Amplitude of terahertz equivalent surface photocurrent versus thickness
of the LTG-GaAs film for different absorption coefficients.
74
55
20
0.5 V/µm
50
45
0
x
E0 = 1 V/µm
10
20
40
60
80
100
120
140
160
K (×106 rad/m)
Photocurrent phase, φ (deg)
Beat frequency = 1 THz
0
180
Angle between two laser beams, θ (deg)
Figure 4.10: Longitudinal component of grating vector, Kx , and phase of terahertz
photocurrent, φ, versus angle between the two laser beams.
75
Chapter 5
Integrated Photomixer-Antenna
Elements
5.1
Introduction
In the last two chapters, we showed that the generated terahertz photocurrents
in superconductive and photoconductive photomixers are travelling wave currents
with a spatiotemporal distribution over the photomixing film. In this chapter we
propose an integrated photomixer-antenna structure as an efficient terahertz source
[81], [82]. Unlike conventional photomixers, in which a generated signal in an active
region is coupled to an external antenna for radiation, in a photomixer-antenna
element, the generated signal inside the photomixing film radiates simultaneously
by designing the film as an efficient radiator. Incorporating a photomixing medium
as an antenna element not only eliminates any source-to-antenna coupling problem,
but also distributes the optical power over a bigger area and hence increases optical
power handling capability of the device.
76
z
1
θ
La
se
r
2
o
Ph
r
se
La
c
du
on
toc ilm
f
as
Bi rod e
ct
to r
el e
w
d
x
l
ǫr
h
+ V -
Figure 5.1: Longitudinal photoconductive photomixer scheme.
5.2
Longitudinal and Transversal PhotomixerAntenna Elements
Fig. 5.1 shows the configuration of a longitudinal photomixer. The two laser beams
shine the entire film under the angle of θ in the (x − z) plane, while the dc bias is
applied in the x direction. In this case the grating vector K is in the (x − z) plane
and hence from Eq. (4.64) the generated terahertz photocurrent can be written as
2
2 1/2 −j(Kx x+φ)
J˜s (x) = [Js1
+ Js2
] e
(5.1)
where Js1 , Js2 , and φ are given by Eqs. (4.67) to (4.69), respectively, and Kx =
(k1 + k2 ) sin θ/2 is the x-component of the grating vector, K.
In this analysis we have not considered the effect of the finite width of the photoconductor film. For more accurate analysis, one has to apply the appropriate
77
z
Laser 2
o
Ph
θ
Laser 1
c
du
on
toc ilm
f
to r
s
Bia rod e
t
c
e
el
w
d
x
l
ǫr
h
+ V -
Figure 5.2: Transversal photoconductive photomixer scheme.
boundary conditions at the edges of the film to take this effect into account.
A transversal photomixer scheme is shown in Fig. 5.2. In this configuration, the
optical interference grating is formed along the width of the photoconductor film,
while the dc bias voltage is applied along the length of the film. In this case, the
grating vector K is in (y − z) plane and hence from Eq. (4.64) the x component of
the photocurrent can be written as
2
2 1/2 −j(Ky y+φ)
J˜s (y) = [Js1
+ Js2
] e
(5.2)
where Js1 , Js2 , and φ are given by Eqs. (4.67) to (4.69), respectively, in which Kx
is replaced by zero, and Ky = (k1 + k2 ) sin θ/2 is the y-component of the grating
vector, K.
Note that in this case the photocurrent has also a diffusion current component in
y direction, which is much smaller than the drift current component given by Eq.
(5.2).
The resulting traveling wave photocurrent distribution given by Eqs. (5.1) and
78
(5.2) can be either guided to a transmission line for circuit applications or can be
radiated out by proper designing of the dimensions of the film.
5.2.1
Radiation from Photomixer-Antenna Elements
Since the current distribution inside the film is known and acts as a source of radiation, the electric current model can be employed to obtain the far field radiation
from the antenna [203], [204].
The radiated field from an x-directed surface current source, J˜s (x, y) on a grounded
substrate can be expressed as
Eθ(φ) (r, θ, φ) =
hex
Eθ(φ)
Z
w/2
−w/2
Z
l/2
J˜s (x, y)ejko sin θ[x cos φ+y sin φ] dxdy
(5.3)
−l/2
hex
where Eθ(φ)
is the radiated electric field from an x-directed Hertzian electric dipole
on a grounded dielectric substrate given by [204]
Eθhex = −E0 G(θ) cos φ
(5.4)
Eφhex = E0 F (θ) sin φ
(5.5)
with
G(θ) =
2 tan[k0 hN (θ)] cos θ
r
cos θ
tan[k0 hN (θ)] − j Nǫ(θ)
2 tan[k0 hN (θ)]
tan[k0 hN (θ)] − jN (θ) sec θ
p
N (θ) = ǫr − sin2 θ
F (θ) =
79
(5.6)
(5.7)
(5.8)
and
E0 =
jΩµ0 −jk0 r
e
4πr
(5.9)
Using Poynting’s vector theorem [205], the radiated power can be written as
Pr =
60
2
2
ξ(Js1
+ Js2
)
π
(5.10)
where ξ is a configuration factor. For the longitudinal case
ξ=
2π
Z
0
π/2
Z
0
sin2 (k0 w/2 sin θ sin φ) sin2 [l/2(k0 sin θ cos φ − Kx )]
·
sin θ
(ko sin θ cos φ − Kx )2
£
¤
× |G(θ)|2 cot2 φ + |F (θ)|2 dθdφ
(5.11)
and for the transversal case
ξ=
Z
0
2π
Z
0
π/2
sin2 (k0 l/2 sin θ cos φ) sin2 [w/2(k0 sin θ sin φ − Ky )]
·
sin θ
(k0 sin θ sin φ − Ky )2
£
¤
× |G(θ)|2 + |F (θ)|2 tan2 φ dθdφ
(5.12)
in which ǫr and h are the relative dielectric constant and the thickness of the
substrate, respectively. The radiated power is maximum for l = π/Kx in the
longitudinal case and for w = π/Ky in the transversal case, at which a resonant
condition occurs for the antenna.
The thickness of the substrate is determined based on the resonant condition and
loss mechanisms of the structure. The best value for the thickness of the substrate,
h, to minimize surface mode loss is [206]
h=
πc
√
2Ω ǫr − 1
80
(5.13)
The width of the photoconductor film in the longitudinal case and the length of
the film in the transversal case determine the level of the absorbed optical power
in the film as well as the directivity of the antenna.
5.2.2
Simulation Results for an LTG-GaAs PhotomixerAntenna Element
A photoconductive photomixer-antenna device made of LTG-GaAs is considered.
All the physical parameters for a typical LTG-GaAs material are presented in Table
4.1.
The central wavelength of one of the lasers is 780 nm. The other laser is a tunable
laser, which its wavelength can be tuned around 780 nm to result in the desired terahertz frequency. The total optical power density of the two lasers is 0.4 mW/µm2 .
The angle between the two laser beams is θ = 4◦ . The resulting grating vector in
the x(y) direction is Kx(y) = 0.561 (µm)−1 . The device is designed to operate at 2
THz. The substrate is GaAs with the relative permittivity of 12.6.
Based on the design criteria explained in the previous section, the optimum values
of the geometrical parameters for the longitudinal and the transversal structures
are computed. For the longitudinal case, the length of the film is l = 5.6 µm, while
the width of the film is w = 50 µm. For the transversal case, the length and the
width of the film are l = 50 µm and w = 5.6 µm, respectively. The thickness of the
film for both cases is d = 2 µm. The thickness of the substrate is a function of the
beat frequency, and hence has a different value at different frequencies. This makes
the proposed devices very narrow band radiators. At 2 THz, the thickness of the
substrate is h = 10.917 µm.
Fig. 5.3 shows the amplitude of the terahertz photocurrent versus beat frequency,
81
14
Longitudinal (VDC=22.4 V)
2
Photocurrent, Js, (A / m )
12
Transversal (VDC=200 V)
10
edηG0 (vsn + vsp )/Ω
8
6
4
2
0
0
0.5
1
1.5
2
2.5
3
Beat frequency (THz)
Figure 5.3: Amplitude of equivalent surface photocurrent in LTG-GaAs longitudinal
and transversal photomixers. The total optical power density of the two lasers is
0.4 mW/µm2 . The applied dc bias to each element is 4 V/µm.
82
which decreases with frequency as 1/Ω at high frequencies. As it can be seen from
Fig. 5.3, the amplitude of the photocurrent is the same for the two cases. This
means that the diffusion current is negligible in these devices, and the drift current
is the main source of the terahertz signal. The dot-line curve shows an approximate expression for the terahertz photocurrent as |J˜s | ≈ edηG0 (vsn + vsp )/Ω, which
predicts accurate values for the terahertz photocurrent at high frequencies (see Eq.
(4.63)). This approximation is not accurate for the low frequency range, where
Ωτ < 1.
Shown in Fig. 5.4 is radiation power versus beat frequency for the transversal
and the longitudinal photomixer-antenna devices. The absorbed optical power in
each device is 74.6 mW. Since the amplitudes of the photocurrents for the two
configurations are the same, the difference in the radiation power is the result of
the difference in their configuration factors ξ (see Fig. 5.5). For the longitudinal
case the photocurrent is a traveling wave current with its phase variation along x
axis, while for the transversal case the phase variation of the traveling-wave photocurrent is along y axis. This results in different configuration factors for the two
cases. From Fig. 5.4, the transversal scheme shows higher radiation power at high
frequencies compare to its longitudinal counterpart; although it needs higher dc
bias voltage due to its longer photo-excited length, l.
There is a frequency range around 2 THz, where two devices show better performance in terms of the radiation power. This behavior can be explained by Figs.
5.3 and 5.5. At low frequency range (< 1 THz), the amplitude of the photocurrent
decreases smoothly with the beat frequency, while the configuration factor increases
by the factor of Ω2 . Hence from Eq. (5.10), the radiation power will increase with
frequency for the low frequency range. At the high frequency range, the photocurrent decreases by the factor of 1/Ω, while the configuration factor linearly increases
83
2
1.8
Tra
1.4
nsv
ers
r
Radiation power, P , (nW)
1.6
1.2
1
Lon
gitu
0.8
al
din
al
0.6
0.4
0.2
0
0
2
4
6
Beat frequency (THz)
8
10
Figure 5.4: Radiation power of LTG-GaAs longitudinal and transversal photomixers. The total power density of the two lasers is 0.4 mW/µm2 . For the longitudinal
case, the length of the film is l = 5.6 µm, while the width of the film is w = 50 µm.
For the transversal case, the length and the width of the film are l = 50 µm and
w = 5.6 µm, respectively. The thickness of the film for both cases is d = 2 µm.
The thickness of the substrate is changed according to the beat frequency to have
maximum radiation at each frequency. The applied dc bias to each element is 4
V/µm.
84
∝Ω
0
9
Configuration factor, ξ, (× 10 )
10
−1
10
2
−2
10
∝
Ω
−3
Longitudinal
10
Transversal
−4
10
0.1
1
Beat frequency (THz)
10
Figure 5.5: Configuration factor, ξ, for longitudinal and transversal photomixerantenna devices.
85
by the beat frequency. This results in a decreasing radiation power by the factor
of 1/Ω at high frequency range.
Radiation power versus beat frequency for the longitudinal photomixer-antenna device operating at 2 THz is shown in Fig. 5.6. Since the structure is designed to
resonate at the frequency of 2 THz (h = 10.917 µm), the maximum radiation occurs at this frequency; although there are other harmonics that the device radiates
reasonable power. One may use this narrow band characteristic of the proposed
device to reduce the effect of the finite linewidth of the optical sources on device
performance.
For the applications where a wideband radiator is needed, bow-tie or spiral antenna
configurations can be used instead of simple rectangular antenna structure. The
other choice is using thick substrate lens instead of the grounded substrate.
86
1.8
r
Radiation power, P , (nW)
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
Beat frequency (THz)
8
10
Figure 5.6: Radiation power of LTG-GaAs longitudinal photomixer designed for 2
THz beat frequency. The length of the film is l = 5.6 µm and the width of the film
is w = 50 µm. The thickness of the substrate is fixed at h = 10.917 µm to have
maximum radiation at 2 THz. The applied dc bias to each element is 4 V/µm.
87
Chapter 6
Photomixer-Antenna Array
The maximum available terahertz power from a single element photoconductive
photomixer source is limited by the maximum sustainable optical power and dc
bias before device failure. The available terahertz power can be increased by using
an array of photomixers. The simple optical excitation scheme of a photomixerantenna element makes it suitable for an array structure. Since there is no need for
a feeding network for an array of these elements, one can excite the entire array by
two laser beams. One can increase the radiation power by increasing the number
of array elements and using high power lasers. On the other hand, one can change
the phase distribution of the photocurrent in array elements by changing the angle
between the two laser beams and steer the radiation beam.
6.1
Photoconductive Photomixer-Antenna Array
In this section, we introduce an array of photoconductive photomixer-antenna elements as a high power tunable CW terahertz source. The radiation characteristics
88
z
Laser 1
(xl1 , yl1 , zl )
knm1
Laser 2
θnm1 θnm2
Photomixer/antenna
element
(xl2 , yl2 , zl )
knm2
Electrode
x
d
ǫr
h
(a)
Laser 1
Photomixer/
antenna array
Laser 2
Cylindrical Lens
Mirror
Mirror
(b)
Figure 6.1: (a) Photoconductive photomixer-antenna array configuration (b) Optical excitation scheme.
89
of the proposed structure is analyzed and the simulation results for a designed sample are presented.
Fig. 6.1 shows the schematic of an (2N + 1) × (2M + 1) array of photoconductive
photomixer-antenna elements. Photomixer elements are made of an ultra-fast photoconductor material, and are biased to a dc voltage by their bias electrodes. All
the array elements are located above a grounded dielectric substrate. Two linearly
polarized laser beams excite the entire structure. The corresponding electric field
associated with each laser over the nm-th element can be written as
Enm1(2) = |Enm1(2) | ej (ω1(2) t−knm1(2) ·r)
(6.1)
where n = −N, ..., N , m = −M, ..., M , ω1 and ω2 are the angular frequencies of the
lasers, knm1 and knm2 are the wave-vectors of the lasers over the nm-th element,
and r is the position vector over the same element. The lasers are assumed to
be far enough from the surface of the photomixers so that the wave reaching each
element can be assumed to be a local plane-wave. A uniform optical excitation can
be achieved by coupling the laser beams to the array via an appropriate cylindrical
lens.
Fig. 6.2 shows the array arrangement in x − y plane. The wave-vectors knm1 and
knm2 can be written as
h
knm1(2) = k1(2) cos βnm1(2) x̂ + cos γnm1(2) ŷ + cos θnm1(2) ẑ
i
(6.2)
where k1 and k2 are the wave-numbers of the lasers in free space and
cos βnm1(2) =
xnm − xl1(2)
[(xnm − xl1(2) )2 + (ynm − yl1(2) )2 + zl2 ]1/2
90
(6.3)
y
+Vdc
+Vdc
+Vdc
dnm
lnm
(xnm ,ynm )
ρnm
d′nm
x
φ′nm
+Vdc
wnm
+Vdc
+Vdc
Figure 6.2: Array arrangement in (x − y) plane.
ynm − yl1(2)
[(xnm − xl1(2) )2 + (ynm − yl1(2) )2 + zl2 ]1/2
−zl
=
[(xnm − xl1(2) )2 + (ynm − yl1(2) )2 + zl2 ]1/2
cos γnm1(2) =
(6.4)
cos θnm1(2)
(6.5)
The incident laser beams interfere inside the array elements and create a spatiotemporal optical power density grating. Considering nonuniform absorption along the
thickness of the photoconducting film, one can write the optical power density
inside the nm-th element as
h
i
Inm (r, t) = I0 Tnm (θnm1 , θnm2 ) 1 + ηnm cos(Ωt − Knm · r) eαz
91
(6.6)
where I0 is the total optical power density of the lasers, Tnm is the optical power
density transmission coefficient at photoconductor-air interface, α is the optical
absorption coefficient, 0 < ηnm ≤ 1 is the modulation index or grating contrast,
Ω = ω1 − ω2 is the angular beat frequency, and Knm = k1nm − k2nm is the grating
vector.
The generated terahertz photocurrent inside the nm-th element is
s
s
J˜nm
(x, y) = Jnm
e−j[Knmx x+Knmy y+φnm ]
(6.7)
Knmx = k1 cos βnm1 − k2 cos βnm2
(6.8)
Knmy = k1 cos γnm1 − k2 cos γnm2
h
i1/2
s
s
s
Jnm
= (Jnm1
)2 + (Jnm2
)2
(6.9)
where
φnm = tan−1
(6.10)
s
Jnm2
s
Jnm1
(6.11)
s
s
The expressions for J1nm
and J2nm
can be extracted from Eqs. (4.67) and (4.68),
respectively.
The far field radiation from the nm-th element can be written as (see Fig. 6.2 for
the definition of the parameters)
hex ˜
nm
(r, θ, φ) = 4Eθ(φ)
Jsnm (xnm , ynm )ejk0 ρnm sin θ cos(φ−φnm )
Eθ(φ)
′
sin[lnm /2(k0 sin θ cos φ − Knmx )]
Knmx − k0 sin θ cos φ
sin[wnm /2(k0 sin θ sin φ − Knmy )]
×
k0 sin θ sin φ − Knmy
×
92
(6.12)
hex
where Eθ(φ)
are given by Eqs. (5.4) to (5.9) [page 79]. The superposition of all
these radiated fields is the field produced by the array
Eθ(φ) (r, θ, φ) =
N
X
M
X
nm
Eθ(φ)
(r, θ, φ)
(6.13)
n=−N m=−M
The total radiated power can be calculated using Poynting’s vector theorem [205]
1
Pr =
2η0
Z
π/2
0
Z
2π
0
[|Eθ |2 + |Eφ |2 ]r2 sin θ dφdθ
(6.14)
where η0 is the free space impedance.
6.1.1
Array Design Procedure
The primary goal is to achieve maximum radiation power, which from Eq. (6.12)
for the nm-th element occurs for
lnm = π/Knmx = λnmx /2
(6.15)
wnm = π/Knmy = λnmy /2
(6.16)
where λnmx and λnmy are the photocurrent spatial wavelengths along x-axis and yaxis, respectively. It can be seen from Eq. (6.8) that the x-component of the grating
vector, Knmx , is a relatively big number. From Eq. (6.15), the optimum values for
the lengths of the array elements will be in the order of a few micrometers. On the
other hand, the y-component of the grating vector, Knmy , given by Eq. (6.9) is a
small number and becomes zero for the array elements located on x-axis. Hence
from Eq. (6.16), the wider the array elements the higher the radiation power.
In Eq. (6.15) both lnm and Knmx are related to the position of the nm-th element
93
in x-y plane. To find the optimum values of the lengths of the array elements from
Eq. (6.15), one has to solve the following iterative equation for xnm
xn(±m) = xn(±(m−1)) ± 1/2ln(±(m−1))
h
(xn(±m) − xl1 )k1
± 3π/2
[(xn(±m) − xl1 )2 + (ynm − yl1 )2 + zl2 ]1/2
i−1
(xn(±m) − xl2 )k2
−
[(xn(±m) − xl2 )2 + (ynm − yl2 )2 + zl2 ]1/2
(6.17)
where m = 1, ..., M , n = −N, ..., N , xn0 = 0, and
ln0 = π
h
i−1
−xl1
xl2
k
+
k
1
2
[x2l1 + (yn0 − yl1 )2 + zl2 ]1/2
[x2l2 + (yn0 − yl2 )2 + zl2 ]1/2
(6.18)
The lengths of the array elements, lnm , and their separation distances in x direction, dnm , are equal to have a constructive spatial power combination, and can be
calculated as
xnm − xl1
k1
[(xnm − xl1 + (ynm − yl1 )2 + zl2 ]1/2
i−1
xnm − xl2
−
k2
[(xnm − xl2 )2 + (ynm − yl2 )2 + zl2 ]1/2
lnm = dnm = π
h
)2
(6.19)
where n = −N, ..., N and m = −M, ..., −1, 1, ..., M .
The thickness of the substrate is determined based on the resonant condition and
loss mechanisms of the structure and is given by Eq. (5.13) [page 80].
This design procedure guarantees maximum radiation for each of the array elements,
but does not necessarily guarantee the maximum radiation for the entire array. To
have maximum radiation for the entire array, one has to maximize Eq. (6.14),
which is difficult to do analytically. As we will see in the next section, the radiation
power for a designed array is close to the maximum radiation of the entire array.
94
6.1.2
Simulation Results for an LTG-GaAs PhotomixerAntenna Array
An 1 × 31 array configuration is considered for simulation purpose. Since the
separation distance of the array elements in y direction is arbitrary, we set it equal
to zero to maximize the photo-excited region and increase the radiation power. The
array elements are made of LTG-GaAs located above a grounded GaAs substrate
with the relative permittivity of 12.6. The elements are biased to a dc voltage of
7.5 V. Note that the applied electric field must be lower than GaAs breakdown
field, which is around 40 V/µm [190], [192]. The thickness of the array elements is
2 µm. All the physical parameters for a typical LTG-GaAs material are presented
in Table 3.1.
The entire array is excited by two detuned single-mode lasers operating around
780 nm with their frequency difference falling in the terahertz spectrum. The
total optical power density of the two lasers is 0.12 mW/µm2 . The LTG-GaAs
photomixers can withstand an optical power density of 0.9 mW/µm2 with 4 V/µm
dc bias at room temperature [46]. The two lasers are located in x − z plane with
xl2 = −xl1 = 1.3 mm and zl = 50 mm. The array is designed to work at 1 THz.
For this frequency the optimum value for the thickness of the substrate is 21.8 µm.
The lengths of the array elements and their separation distances are calculated from
Eqs. (6.17) to (6.19), and are around 7.46 µm. The width of the array elements is
120 µm.
Fig. 6.3 shows radiation power versus beat frequency for the designed array. The
power of each laser is 3.33 W, and the consumed optical power in each element is
107 mW. A Ti:sapphire laser can generate the required optical power. In order to
decrease the thermal effects of the absorbed optical power and the applied dc bias,
95
1.4
1.2
Radiation power ( µW )
1
0.8
0.6
0.4
0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Beat frequency (THz)
Figure 6.3: Total terahertz radiation power of an 1 × 31 photoconductive
photomixer- antenna array made of LTG-GaAs versus beat frequency. The thickness of the substrate is fixed at 21.8 µm to have maximum radiation at 1 THz. The
total optical power density of the two lasers is 0.12 mW/µm2 . The lengths of the
array elements and their separation distances are calculated from Eqs. (6.17) to
(6.19), and are around 7.46 µm. The width of the array elements is 120 µm. The
elements are biased to a dc voltage of 7.5 V.
96
10
Radiation power ( µW )
1
0.1
f = 0.1 THz
0.01
f = 0.5 THz
f = 1 THz
0.001
0.1
f = 2 THz
1
10
Applied electric field, E0 (V/µm)
Figure 6.4: Total terahertz radiation power of an 1 × 31 photoconductive
photomixer-antenna array made of LTG-GaAs versus applied electric field at different beat frequencies. The thickness of the substrate is changed according to the
beat frequency to have maximum radiation at each frequency. The total power
density of the two lasers is 0.12 mW/µm2 .
97
one can use a good thermal conductor layer, such as AlAs layer [78], beneath the
LTG-GaAs layer. Operating the photomixer at the low temperatures is another way
to increase the maximum sustainable optical power and dc bias of the device [46].
It can be seen from Fig. 6.3 that around 1 µW power is achievable at 1 THz. The
optical-to-electrical efficiency of the device is 0.0032%. The optimum thickness of
the substrate is a function of frequency, which makes the structure a narrow band
radiator.
Shown in Fig. 6.4 is radiated power versus applied dc bias. The applied electric
field is the ratio of the applied dc voltage to the length of the array elements. The
radiation power saturates with the applied electric field, which is the result of the
carrier velocity saturation in photoconductor. As it can be seen from Fig. 6.4, for an
LTG-GaAs sample with the physical parameters given in Table 3.1, the saturation
region starts at the applied electric field around E0 = 3 V/µm for the beat frequency
above 0.5 THz. The optimum value of the applied dc bias for maximum terahertz
radiation and minimum thermal effect is the value corresponding to the starting
point of the saturation region.
Fig. 6.5 shows radiation power versus beat frequency, where the thickness of the
substrate is changed according to the beat frequency to have maximum radiation at
each frequency. According to Fig. 6.5, the radiation power decreases with frequency
by the rate of 1/Ω [79].
From Fig. 6.1, the angle between the beams of the two lasers located in x − z
plane can be defined as θl = θ001 + θ002 = − tan−1 (xl1 /zl ) + tan−1 (xl2 /zl ). Fig.
6.6 shows radiation pattern at the plane of φ = 0 for different values of θl , and
for arrays with different number of elements. It can be seen that by changing the
angle between the two lasers one can steer the radiation beam. The grating vector,
Knm , changes by changing the angle between the two lasers, which consequently
98
3
Radiation power ( µW )
2.5
E0 = 2.5 V/µm
2
1.5
1
0.5
0
1
2
3
4
5
Beat frequency (THz)
Figure 6.5: Total terahertz radiation power of an 1 × 31 photoconductive
photomixer-antenna array made of LTG-GaAs versus beat frequency. The thickness of the substrate changes according to the beat frequency to have maximum
radiation at each frequency. The total optical power density of the two lasers is
0.12 mW/µm2 .
99
θ
θ
o
o
o
o
90o
o
60o
60
60
90
30
o
o
60
o
30
30
30
90o
90o
θl = 3o
θl = 2.9o
(a)
o
θl = 3.1o
θ
o
30
o
o
60
60
o
o
o
90
30o
o
o
60
90
θ
30o
30
60o
(b)
90
90
(d)
(c)
Figure 6.6: Radiation pattern at φ = 0 plane for different number of array elements
and at the beat frequency of 1 THz; (a) 1×31 array (FWHM=6.59o ), (b) 1×51 array
(FWHM=4o ), (c) 1 × 71 array (FWHM=2.88o ), (d) 1 × 91 array (FWHM=2.3o ).
100
changes the phase distribution of the current over the array elements and results
in pattern deflection. It also can be seen from Fig. 6.6 that the radiation beam
becomes narrower by increasing the number of array elements. Fig. 6.7 shows the
radiation pattern of the designed array at φ = 0 and φ = π/2 planes. The radiation
pattern is symmetric and its side-lobe level is around -13.2 dB.
Shown in Fig. 6.8 is radiation power for different values of the angle between the
two lasers. As it can be seen from Fig. 6.8, the maximum radiation of the entire
array occurs at an angle very close to θl = 3o . The radiation power rapidly drops
for the angles far from the angle that the array is designed for, because at these
angles the maximum radiation condition given by Eq. (6.15) is no longer valid. The
line shown in Fig. 6.8 is asymmetric with respect to θl , since the grating vector
Knm is not symmetric with respect to θl .
Fig. 6.9 shows that one can increase the terahertz radiation power by using higher
power lasers and increasing the number of array elements. The radiation power does
not scale as the square of the number of array elements, while it is proportional to
the square of the optical power (see Fig. 6.10). Hence, for a given amount of the
optical power, the optimum design is an array with minimum number of elements,
while each element absorbs its maximum consumable optical power.
101
0
−10
Relative power (dB)
−20
−30
−40
−50
−60
−70
−80
φ=0
−90
φ=π/2
−90
−70
−50
−30
−10
10
30
50
70
90
θ (deg)
Figure 6.7: Radiation pattern at φ = 0 and φ = π/2 planes for an 1 × 31 array at
the beat frequency of 1 THz and θl = 3.02o .
102
1.4
1.2
Radiation power ( µW )
1
0.8
0.6
0.4
0.2
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
Angle between two lasers ( deg )
Figure 6.8: Terahertz radiation power versus angle between the two lasers at the
beat frequency of 1 THz.
103
2.8
2.6
Radiation power ( µW )
2.4
2.2
2
1.8
1.6
1.4
1.2
31
41
51
61
71
81
91
Number of array elements
Figure 6.9: Terahertz radiation power versus number of array elements at the beat
frequency of 1 THz and for a constant optical power density of 0.12 mW/µm2 .
104
10
9
Radiation power (µW)
8
Number of array elements= 7
E0 = 1 V/µm
7
6
5
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2
Optical power density (mW / µm )
Figure 6.10: Terahertz radiation power versus total optical power density at the
beat frequency of 1 THz.
105
z
Laser 1
Laser 2
(x1l , zl )
(x2l , zl )
k1i
k2i
θ1i
HTS film
θ2i
Electrode
Idc
xi
x0
d
h
di
xN
x
w
li
ǫr
Ground Plane
Figure 6.11: Superconductive photomixer-antenna array configuration.
6.2
Superconductive Photomixer-Antenna Array
Fig. 6.11 shows the schematic of an array of HTS photomixer-antenna elements.
The array structure consists of an epitaxially grown HTS thin film on a suitable
substrate using a growing technique such as molecular beam epitaxy (MBE), sputtering, laser ablation, or metal-organic chemical vapor deposition (MOCVD) [207].
The HTS film can be patterned using standard photolithographic techniques. The
electrodes can be printed on the HTS film by lift-off techniques followed by an annealing process in an oxygenated environment to reduce the contact resistance and
improve the adhesion of the metal pads to the HTS film [177]. The ground plane
can be deposited on the back side of the substrate by evaporation or sputtering
techniques. A dc current source connected to the electrodes provides a constant
bias current for the array elements. One can change the number of array elements
by connecting the current source to the appropriate electrode.
Two linearly polarized lasers are located in x − z plane far enough from the surface
106
of the photomixers so that the wave reaching each element can be assumed to be
a local plane-wave. The corresponding electric field associated with each laser over
the i-th element can be written as
E1(2)i = |E1(2)i | ej (ω1(2) t−k1(2)i ·r)
(6.20)
where i = −N, ..., N , ω1 and ω2 are the angular frequencies of the lasers, k1i and
k2i are the wave-vectors of the lasers over the i-th element, and r is the position
vector over the same element.
The wave-vectors k1i and k2i can be written as
(xi − x1l )x̂ − zl ẑ
k1
[(xi − x1l )2 + zl2 ]1/2
(xi − x2l )x̂ − zl ẑ
k2
k2i =
[(xi − x2l )2 + zl2 ]1/2
k1i =
(6.21)
(6.22)
where k1 and k2 are the wave-numbers of the lasers in free space.
The incident laser beams interfere inside the elements and create a spatiotemporal
grating. The optical power distribution inside the i-th element can be written as
h
i
Pi (r, t) = P0i (1 − R) 1 + ηi cos(Ωt − Ki · r) eαz
(6.23)
where P0i is the total power of the lasers, R is the optical reflectivity, α is the optical
absorption coefficient, 0 < ηi ≤ 1 is the modulation index or grating contrast,
Ω = ω1 − ω2 is the angular beat frequency, and Ki = k1i − k2i is the grating vector.
The total equivalent surface current density over the i-th element of the array can
be written as [80]
s
Jis (x, t) = Jdc
+ ℜe{J˜is ejΩt }
107
(6.24)
where
s
Jdc
=
Idc
w
J˜is = Jis e−j[Kix x+φi ]
(6.25)
(6.26)
with
s 2
s 2 1/2
Jis = [(J1i
) + (J2i
)]
(xi − x2l )k2
(xi − x1l )k1
−
2 1/2
2
[(xi − x1l ) + zl ]
[(xi − x2l )2 + zl2 ]1/2
Js
φi = tan−1 2i
s
J1i
Kix =
(6.27)
(6.28)
(6.29)
s
s
The expressions for J1i
and J2i
can be extracted from Eqs. (3.40) and (3.41),
respectively.
As it can be seen from Eq. (6.24), the total current inside each element contains a
dc component and a traveling-wave terahertz component, where the dc component
is equal to the bias current. From energy conversion point of view, in each element
a part of the energy of the exciting lasers is converted to the terahertz signal [80].
Having the current distribution over each element, the far field radiation can be
calculated as
i
Eθ(φ)
(r, θ, φ)
=
hex
Eθ(φ)
Z
w/2
−w/2
Z
xi +li /2
xi −li /2
Jis e−j[Kix x+φi ] × ejk0 sin θ[x cos φ+y sin φ] dxdy (6.30)
Doing the integration results in
i
hex s −jφi j[k0 sin θ cos φ−Kix ]xi
Eθ(φ)
(r, θ, φ) = 4Eθ(φ)
Ji e
e
×
sin[li /2(k0 sin θ cos φ − Kix )] sin[w/2(k0 sin θ sin φ)]
Kix − k0 sin θ cos φ
k0 sin θ sin φ
108
(6.31)
hex
where Eθ(φ)
are given by Eqs. (5.4) to (5.9) [page 79] and k0 is the free space wave-
number. The superposition of all these radiated fields is the field produced by the
array
Eθ(φ) (r, θ, φ) =
N
X
i
Eθ(φ)
(r, θ, φ)
(6.32)
i=−N
The total radiated power can be calculated using Poynting’s vector theorem [205]
1
Pr =
2η0
Z
0
π/2
Z
2π
0
[|Eθ |2 + |Eφ |2 ]r2 sin θ dφdθ
(6.33)
where η0 is the free space impedance.
6.2.1
Array Design Procedure
The length and the location of the array elements must be chosen judiciously to
have maximum radiation for each element and constructive power combination.
From Eq. (6.31), the maximum radiation for the i-th element occurs for
li = π/Kix = λix /2
(6.34)
where λix is the photocurrent spatial wavelength along x-axis.
In Eq. (6.34), both li and Kix are related to the position of the i-th element along
x-axis. To find the optimum values for the lengths of the array elements from Eq.
(6.34), one has to solve the following iterative equation for xi
1
x±i =x±(i−1) ± l±(i−1)
2
i−1
(x±i − x2l )k2
(x±i − x1l )k1
3π h
−
±
2 [(x±i − x1l )2 + zl2 ]1/2 [(x±i − x2l )2 + zl2 ]1/2
109
(6.35)
where i = 1, ..., N , x0 = 0, and
i−1
−x1l
x2l
l0 = π
k1 + 2
k2
(x21l + zl2 )1/2
(x2l + zl2 )1/2
h
(6.36)
To have a constructive spatial power combining, one has to choose di = li . Once xi
is calculated from Eq. (6.35), one can calculate the lengths of the array elements li
and their separation distances di as
li = di = π
h
i−1
(xi − x2l )k2
(xi − x1l )k1
−
[(xi − x1l )2 + zl2 ]1/2 [(xi − x2l )2 + zl2 ]1/2
(6.37)
where i = −N, ... − 1, 1, ..., N .
The thickness of the substrate is given by Eq. (5.13) [page 80].
To have maximum radiation for the entire array, one has to maximize the expression
given by Eq. (6.33), which is difficult to do analytically. As we will see in the
next section, after designing the array based on the above-mentioned procedure,
the maximum radiation can be achieved by a small tuning of the positions of the
lasers.
6.2.2
Simulation Results for an YBCO Photomixer-Antenna
Array
An array with 61 elements is considered for simulation purpose. The array elements
are made of YBa2 Cu3 O7−δ high-temperature superconductor, which is epitaxially
grown on LaAlO3 substrate with relative permittivity of 24. The ground plane is
deposited on the back side of the substrate. The thickness and the width of the
array elements are 130 nm and 40 µm, respectively. The array elements are biased
to a dc current of 23.8 mA. All the physical parameters for a typical YBa2 Cu3 O7−δ
110
material are presented in Table 4.1.
The entire array is excited by two detuned single-mode lasers operating around 532
nm with their frequency difference falling in the terahertz spectrum. Each laser
radiates 114 mW optical power, which results in 3.4 mW total absorbed power for
each element. The two lasers are located in x − z plane with x2l = −x1l = 1.3 mm
and zl = 50 mm.
From Fig. 6.11, the angle between the two lasers can be defined as θl = θ10 + θ20 =
− tan−1 (x1l /zl ) + tan−1 (x2l /zl ), which results in θl = 3o for this specific configuration. The array is designed to work at 3 THz frequency.
At this frequency, the
optimum value for the thickness of the substrate is 5.21 µm. The lengths of the
array elements and their separation distances are calculated from Eqs. (6.35) to
(6.37) (see Fig. 6.12).
Shown in Fig. 6.13 is radiation power versus angle between the two lasers. As it
can be seen from Fig. 6.13, the optimum value for the angle between the two lasers
is θl = 3.08o , where the maximum radiation of the entire array happens. Hence,
after designing the array for θl = 3o , the lasers must be relocated to have θl = 3.08o .
The radiation power rapidly drops for the angles far from the angle that the array
is designed for, since at these angles the maximum radiation condition given by Eq.
(6.34) is no longer valid.
Fig. 6.14 shows radiation power versus beat frequency for the designed array. It can
be seen that about 0.2 mW power is achievable at 3 THz. The bath temperature
is T0 = 85 K. Note that upon optical excitation, the temperature of the HTS film
increases above the bath temperature. To have the HTS film in its superconducting
state, the power of the lasers must be smaller than a critical value [80]. Fig. 6.14
also shows the variation of the radiation power with thickness of the substrate.
Fig. 6.15 shows radiation power versus beat frequency at two different bath tem-
111
5.098
5.097
5.095
i
i
l , d (µm)
5.096
5.094
5.093
5.092
−30
−20
−10
0
10
Element number, i
20
30
Figure 6.12: Lengths of the array elements and their separation distances.
112
0.25
Radiation power ( mW )
0.2
0.15
0.1
0.05
0
2.4
2.6
2.8
3
3.2
3.4
3.6
Angle between two lasers ( deg )
Figure 6.13: Terahertz radiation power versus angle between the two lasers.
113
peratures, where the thickness of the substrate is changed according to the beat
frequency to have maximum radiation at each frequency. As it can be seen from
Fig. 6.15, the radiation power increases with the beat frequency, the behavior that
has been explained in Chapter 3, and also has been reported by Stevens and Edwards [57]. Fig. 6.15 also shows that the radiation power is higher for the bath
temperatures closer to the critical temperature of the HTS material. The maximum
obtainable frequency is limited by the gap frequency of the YBCO sample, which
can be from 5 THz to 30 THz for different samples.
Fig. 6.16 shows radiation pattern for different values of θl . The array has a narrow
beam radiation pattern with - 13 dB side-lobe level. It can be seen that by changing the angle between the two lasers one can steer the radiation beam. The beam
steering is due to the change of the grating vector, Ki , which consequently changes
of the current phase distribution over the array elements.
One can increase the terahertz radiation power by using higher power lasers and
increasing the number of array elements. The size of an array with 101 elements
is 1 mm × 40 µm. Using two lasers with total optical power of 379 mW, one can
achieve 0.44 mW power at 3 THz for an array with 101 elements.
114
10.43
7.82
1.5
2
Thickness of substrate (µm)
6.26
4.47
5.21
3.91
3.48
0.25
Radiation power (mW )
0.2
0.15
0.1
0.05
0
1
2.5
3
3.5
4
4.5
5
Beat frequency (THz)
Figure 6.14: Terahertz radiation power versus beat frequency for an YBCO array
with 61 elements designed to operate at 3 THz. The thickness of the substrate
is 5.21 µm. The thickness and the width of the array elements are 130 nm and
40 µm, respectively. The bath temperature is T0 = 85 K. Each laser radiates 114
mW optical power. The applied dc bias is 23.8 mA.
115
1
T =85 K
0
P =229 mW
0
Radiation power ( mW )
0.1
T =82 K
0
P0=572 mW
0.01
0.001
1
1.5
2
2.5
3
3.5
4
Beat frequency (THz)
Figure 6.15: Total terahertz radiation power for an YBCO array with 61 elements versus beat frequency at two bath temperatures. The critical temperature is
Tc = 87 K. The thickness of the substrate changes according to the beat frequency
to have maximum radiation at each frequency. The applied dc bias is 23.8 mA.
116
0
−10
−20
Relative power (dB)
−30
−40
−50
−60
−70
o
θ =3
l
−80
o
θ =3.08
l
−90
o
θ =2.87
l
−100
−110
−80
−60
−40
−20
0
20
40
60
80
θ (deg)
Figure 6.16: Radiation pattern of the array at φ = 0 plane for different angles
between the two lasers.
117
Chapter 7
Photomixer-Antenna Array with
Integrated Excitation Scheme
In the photomixer-antenna array structures introduced in chapter 6 more than %50
of the exciting laser power either reflects back at the photomixer surface or shines
the unwanted areas. Also for these structures, the configuration is very sensitive to
vibration and beam directing setup. In this chapter, we propose a photoconductive
photomixer-antenna array with integrated excitation scheme, in which the laser
beams are guided inside the substrate and excite the photomixer elements. In
this way the laser power is only being consumed by photomixer elements, and the
photomixer-antenna elements can be integrated with other optical components on
a single substrate. Since the excitation of the substrate waveguide can be done by
optical fiber coupling, the whole structure is robust and less sensitive to vibration
and other environmental parameters.
118
Photomixer/
antenna element
Electrode (LTG-GaAs)
nc = 1
z
l′
l
d
hf
x
nf = 3.55
Al 0.1 Ga 0.9 As
Laser 1
Laser 2
Silicon
ns = 3.418
Ground plane
hs
(a)
xi
Ẽ1i (x, z, t)
xi − l/2
Ẽ2i (x, z, t)
xi + l/2
(b)
Figure 7.1: (a) A photomixer-antenna array with integrated excitation scheme. The
laser beams are guided inside the core layer made of Al0.1 Ga0.9 As with refractive
index of nf = 3.55 and excite the photomixer-antenna elements made of LTG-GaAs.
(b) Electric field distribution around i-th element.
7.1
Fig.
Array Design I
7.1 shows the schematic of a photoconductive photomixer-antenna array
with integrated excitation scheme. The photomixer-antenna elements are located
above a grounded dielectric slab waveguide structure. The wavelengths of the
lasers are around 780 nm. The laser beams are guided inside the core layer made
of Al0.1 Ga0.9 As with refractive index of nf = 3.55 and with band gap around 1.5
eV [208]. The photomixer-antenna elements made of LTG-GaAs are buried inside
the core layer. The Al0.1 Ga0.9 As layer is located above a grounded dielectric layer
made of silicon with refractive index of ns = 3.418. Since the refractive index of
119
LTG-GaAs (nGaAs = 3.58) is close to that of Al0.1 Ga0.9 As, we neglect any reflection
from LTG-GaAs/Al0.1 Ga0.9 As interfaces, and we assume that the entire structure
is a three layer asymmetric single mode slab waveguide. Although this is a good
approximation for an array with limited number of elements, however, for an array
with a large number of elements even a small difference in the refractive index of
the waveguide and the photomixer elements may cause an inaccuracy in the results
of the simulation.
The lasers’ beam are lunched inside the core of the waveguide structure from two
sides of the waveguide and excite the photomixer-antenna elements while they travel
along the length of the waveguide. The thickness of the film is designed in such a
way that the waveguide becomes a single mode waveguide at the given lasers’ frequencies. The electric field of the TE0 mode corresponding to the two laser beams
over the i-th element can be written as (see Fig. 7.1)
1
1
1
1
′
Ẽ1i (x, z, t) = E1 (z)e− 2 α(x−xi +l/2) e− 2 (i−1)αl ej[ω1 t−β1x (x+N l +(N +1/2)l]
′
Ẽ2i (x, z, t) = E2 (z)e− 2 α(x−xi −l/2) e− 2 (2N +1−i)αl ej[ω2 t−β2x (x−N l −(N +1/2)l]
xi − l/2 ≤ x ≤ xi + l/2
(7.1)
(7.2)
(7.3)
where α is the absorption coefficient, ω1 and ω2 are the angular frequency of the two
lasers, and β1x and β2x are the wavenumbers of the propagating modes, respectively.
The amplitude of the two modes, E1 (z) and E2 (z) can be written as [209]
E1(2) (z) = A1(2) cos(β1(2)z z − φ)
120
(7.4)
where
i1/2
2ω1(2) µ0 P1(2)
A1(2) =
1
W β1(2)x hf (1 + 2w
+ 2w1 ′ )
£ 2 2
¤1/2
β1(2)z = k1(2)
nf − β1(2)x
¡w¢ 1
¡ w′ ¢
νπ 1
− tan−1
,
+ tan−1
φ=
2
2
u
2
u
h
(7.5)
(7.6)
ν = 0, 1, ...
(7.7)
in which P1(2) are the two lasers’ power, W is the width of the waveguide, k1(2) are
the two lasers’ wavenumber in free space, and
£
¤1/2
β1(2)x = k1(2) n2s + b(n2f − n2s )
v
q
hf =
max{k1 , k2 } n2f − n2s
√
w=v b
p
w′ = v b + γ
√
u=v 1−b
1
νπ
√
tan−1 γ +
,
2
2
n2 − n2c
γ = 2s
nf − n2s
v=
ν = 1, 2, ...
(7.8)
(7.9)
(7.10)
(7.11)
(7.12)
(7.13)
(7.14)
The parameter b is the normalized propagation constant, which can be calculated
from the following dispersion equation [209]
p
p
√
2v 1 − b = νπ + tan−1 b/(1 − b) + tan−1 (b + γ)/(1 − b)
(7.15)
where ν = 0, 1, 2, ... is the mode number.
The optical power density inside the i-th element is proportional to the square of
121
the total electric field and can be written as
ne h 2
Ii (x, z, t) =
E1 (z)e−α
2η0
¡
x−xi +(i−1/2)l
¢
+
¡
E22 (z)eα x−xi −(2N −i+3/2)l
i
2E1 (z)E2 (z)e−α(N +1/2)l cos(Ωt − Kx x − Φ)
¢
(7.16)
where
ne = [n2s + b(n2f − n2s )]1/2
(7.17)
Kx = β1x + β2x
(7.18)
Ω = ω1 − ω2
(7.19)
£
¤
Φ = (β1x − β2x ) N l′ + (N + 1/2)l
(7.20)
The carrier generation rate in terms of the optical power density can be written as
Gi (x, z, t) =
α
Ii (x, z, t)
ℏω
= G1i (x, z) + G2i (x, z) + δG(z) cos(Ωt − Kx x − Φ)
(7.21)
where ω is the average of the angular frequencies of the two lasers and
¡
¢
i
αne 2 −α x−xi +(i−1/2)l h
1 + cos(2β1z z − 2φ)
(7.22)
A1 e
2ℏωη0
¡
¢
i
αne 2 α x−xi −(2N −i+3/2)l h
1 + cos(2β2z z − 2φ)
(7.23)
A2 e
G2i (x, z) =
2ℏωη0
h
¡
¢
¡
¢i
αne
A1 A2 e−α(N +1/2)l cos (β1z + β2z )z − 2φ + cos (β1z − β2z )z
δG(z) =
2ℏωη0
G1i (x, z) =
(7.24)
122
By taking the average of G1i (x, z), G2i (x, z), and δG(z) over the thickness and the
length of the i-th element, one can find the effective carrier generation rate inside
the i-th element as
h
i
Ḡi (x, t) = Ḡ0i 1 + ηi cos(Ωt − Kx x − Φ)
(7.25)
Ḡ0i = Ḡ1i + Ḡ2i
(7.26)
where
ηi =
δ Ḡ
Ḡ1i + Ḡ2i
(7.27)
with
Ḡ1i =
h
i
1
ne A21 −α(i−1)l
e
(1 − e−αl ) d +
sin(β1z d) cos(β1z (2hf − d) − 2φ)
4dlℏωη0
β1z
(7.28)
h
i
1
ne A22 −α(2N +1−i)l
e
(1 − e−αl ) d +
sin(β2z d) cos(β2z (2hf − d) − 2φ)
Ḡ2i =
4dlℏωη0
β2z
(7.29)
"
³β + β
´
αne A1 A2 −α(N +1/2)l
1
β1z + β2z
1z
2z
d cos
(2hf − d) − 2φ
δ Ḡ =
e
sin
dℏωη0
β1z + β2z
2
2
#
³β − β
´
1
β1z − β2z
1z
2z
+
sin
(7.30)
d cos
(2hf − d)
β1z − β2z
2
2
The carrier generation rate given by Eq. (7.25) has the same form as Eq. (4.4),
except for the term eαz . One can use the analysis given in Chapter 4 and find the
generated terahertz photocurrent inside the i-th element as
J˜is (x) = Jis e−j[Kx x+φi ]
123
(7.31)
where
Jis
=
h
s 2
(J1i
)
+
s 2
(J2i
)
φi = Φ + tan−1
s
J2i
s
J1i
i1/2
(7.32)
(7.33)
with
h
s
ˆ i | cos φni + vp |δP
ˆ i | cos φpi
J1i
=ed vn |δN
ˆ i | sin φni − Dp Kx |δP
ˆ i | sin φpi
+ Dn Kx |δN
i
h
s
ˆ i | sin φni − vp |δP
ˆ i | sin φpi
J2i
=ed − vn |δN
ˆ i | cos φni − Dp Kx |δP
ˆ i | cos φpi
+ Dn Kx |δN
ˆ i =|δN
ˆ i |ejφni = ani + jbni ηi Ḡ0i
δN
ai + jbi
ˆ i =|δP
ˆ i |ejφpi = api + jbpi ηi Ḡ0i
δP
ai + jbi
e
−1
an[p]i =τp[n]
+ Dp[n] Kx2 + (µn N̄0i + µp P̄0i )
ǫ
bn[p]i =Ω − µp[n] E0 Kx
e
ai =ani api − bni bpi − [µp ani P̄0i + µn api N̄0i ]
ǫ
e
bi =api bni + ani bpi − [µp bni P̄0i + µn bpi N̄0i ]
ǫ
N̄0i [P̄0i ] =N̄01i [P̄01i ] + N̄02i [P̄02i ]
(7.34)
i
(7.35)
(7.36)
(7.37)
(7.38)
(7.39)
(7.40)
(7.41)
(7.42)
The parameters N̄01i , N̄02i , P̄01i , and P̄02i can be calculated from the equations
given in Appendix A, where G0 is replaced by Ḡ1i to calculate N̄01i and P̄01i , and
is replaced by Ḡ2i to calculate N̄02i and P̄02i .
124
The terahertz photocurrent distribution given by Eq. (7.31) has the same form as
that of Eq. (6.26). The radiation power can be calculated by Eqs. (6.31) to (6.33).
The length of the elements and the separation distance between the elements are
equal and are given by
l = l′ =
7.2
π
Kx
(7.43)
Simulation Results
An array with 19 elements is considered. The physical parameters of the LTGGaAs material are given in Table 3.1. The thickness of the core layer, hf , for single
mode operation at 780 nm is 0.573 µm. The thickness of the substrate, hs , at the
frequency of 1 THz is 22.4 µm. The grating vector, Kx , at the frequency of 1 THz
is 56.5 (µm)−1 . From Eq. (7.43), the length of the array elements at the frequency
of 1 THz is 55.6 nm. The width of the array elements is 100 µm. The dc bias
voltage over each element is 0.139 V. The optical power coupled to the waveguide
from each of the lasers is 100 mW.
Fig.
7.2 shows the radiation power versus beat frequency for different values
of the thickness of the photomixer-antenna elements. The thickness of the substrate is changed according to the beat frequency to have maximum radiation at
each frequency. The radiation power increases by increasing the thickness of the
photomixer-antenna elements.
Fig. 7.3 shows the absorbed optical power in each element for the array elements
with the thickness of 0.573 µm working at 1 THz. The length of the array elements is very small, which limits the maximum consumable optical power in each
element. It also makes it difficult to fabricate the designed array. High value of
the grating vector given by Eq. (7.18) results in a small value for the length of the
125
array elements. In the next section we analyze an array configuration, in which
the two laser beams are lunched from one side of the optical waveguide. In this
configuration, the resulting grating vector is equal to the subtraction of the two
propagation constants of the two beams inside the waveguide, which results in the
resonant length of the array elements in the order of 40 µm.
126
1.4
d=0.573 µm
1.2
Radiation power (nW )
1
d=0.43 µm
0.8
0.6
0.4
d=0.286 µm
0.2
0
0.5
1
1.5
2
2.5
3
Beat frequency (THz)
Figure 7.2: Terahertz radiation power versus beat frequency for different values of
the thickness of the photomixer-antenna elements. The number of array elements is
19. The thickness of the core layer is hf = 0.573 µm. The thickness of the substrate
at the frequency of 1 THz is , hs = 22.4 µm. The length of the array elements at
the frequency of 1 THz is 55.6 nm. The width of the array elements is 100 µm.
The dc bias voltage over each element is 0.139 V. The optical power coupled to the
waveguide from each of the lasers is 100 mW.
127
11.5
11
Absorbed optical power(mW )
10.5
10
9.5
9
8.5
8
7.5
7
0
5
10
Element number, i
15
Figure 7.3: Absorbed optical power in each element.
128
20
Electrode
Photomixer/
antenna element
nc = 1
z
li
li′
d
x
n
f = 3.55
hf
ns = 3.418
hs
Laser 1
and
Laser 2
Ground plane
Figure 7.4: A photomixer-antenna array with integrated excitation scheme: two
laser beams are lunched from one side of the optical waveguide.
7.3
Array Design II
In this section we consider a photomixer-antenna array as shown in Fig. 7.4, where
the two laser beams are lunched from the left side of the waveguide. In this case
the electric field of the TE0 mode corresponding to the two laser beams over the
i-th element can be written as
1
t
Ẽ1(2)i (x, z, t) = E1(2) (z)e− 2 α(x−xi +li +li /2) ej[ω1(2) t−β1(2)x (x+x0 )]
(7.44)
where E1(2) (z) are given by Eq. (7.4), β1(2)x are given by Eq. (7.8), and
lit
=
i−1
X
ln
(7.45)
n=1
x0 =
N
X
(ln + ln′ ) +
n=1
129
lN +1
2
(7.46)
The optical power density inside the i-th element can be written as
i
ne −α(x−xi +lit +li /2) h 2
2
Ii (x, z, t) =
e
E1 (z) + E2 (z) + 2E1 (z)E2 (z) cos(Ωt − Kx x − Φ)
2η0
(7.47)
where
Kx = β1x − β2x
(7.48)
Ω = ω1 − ω2
(7.49)
Φ = (β1x − β2x )x0
(7.50)
The effective carrier generation rate inside the i-th element can be written as
h
i
Ḡi (x, t) = Ḡ0i 1 + ηi cos(Ωt − Kx x − Φ)
where Ḡ0i = Ḡ1i + Ḡ2i and ηi =
ne A21(2)
(7.51)
δ Ḡi
with
Ḡ1i + Ḡ2i
t
e−αli (1 − e−αli )
Ḡ1(2)i =
4dli ℏωη0
h
i
1
d+
sin(β1(2)z d) cos(β1(2)z (2hf − d) − 2φ)
β1(2)z
ne A1 A2 −αlit
e
(1 − e−αli )
δ Ḡi =
dli ℏωη0
"
³β + β ´
³β + β
´
1
1z
2z
1z
2z
sin
d cos
(2hf − d) − 2φ
β1z + β2z
2
2
#
³β − β
´
³β − β ´
1
1z
2z
1z
2z
d cos
(2hf − d)
sin
+
β1z − β2z
2
2
(7.52)
(7.53)
One can calculate the generated terahertz photocurrent inside the i-th element
using Eqs. (7.31)-(7.42) and the radiation power using Eqs. (6.31)-(6.33).
130
For this design, the resonant length of the array elements given by Eq. (7.43) is
in the order of 40 µm. One has to choose the lengths of the array elements in
such a way that the consumed optical power in each element becomes less than the
maximum sustainable power by that element. In other words, the lengths of the
array elements will be much less than the resonant length. We design the lengths
of the array elements to have equal absorbed optical power in each element.
Fig. 7.5 shows the power flow diagram along the array structure. For equal absorbed
optical power inside the i-th and the (i + 1)-th elements one can write
in
out
Piin − Piout = Pi+1
− Pi+1
(7.54)
where
Piout = Piin e−αli
(7.55)
out
in −αli+1
Pi+1
= Pi+1
e
(7.56)
Substituting Eqs. (7.55) and (7.56) into Eq. (7.54) results in
1
li+1 = − ln(2 − eαli ),
α
Eq. (7.57) is valid for li <
1
α
i = 1, 2, ..., 2N + 1
(7.57)
ln 2.
The length of the first element, l1 can be calculated from
(W l1 )Pmax = (1 − e−αl1 )P0
(7.58)
where W is the width of the array elements, Pmax is the maximum sustainable
optical power by the photomixer-antenna elements in W/m2 , and P0 is the total
131
pin
i−1
li−1
pout
i−1
li
i
pin
i
pout
pin
i
i+1
li+1
pout
i+1
Figure 7.5: Power flow diagram along the array structure.
input optical power.
7.4
Simulation Results
An array with 5 elements is designed. The thickness of the core layer, hf , for single
mode operation at 780 nm is 0.573 µm. The thickness of the substrate, hs , at the
frequency of 1 THz is 22.4 µm. The grating vector, Kx , at the frequency of 1
THz is 0.0735 (µm)−1 . The optical power of each laser is 100 mW. Assuming the
maximum sustainable optical power by each element is 0.9 mW/µm2 , the lengths
of the array elements can be calculated from Eqs. (7.57) and (7.58) as 220 nm,
282 nm, 395 nm, 663 nm, and 2.814 µm. The thickness and the width of the array
elements are 0.573 µm and 200 µm, respectively. The absorbed optical in each
element is 39.5 mW. The applied electric field over each element is 2.5 V/µm.
Fig. 7.6 shows radiation power versus beat frequency. The thickness of the substrate
is changed according to the beat frequency to have maximum radiation at each
frequency.
132
4
3.5
Radiation power (nW )
3
2.5
2
1.5
1
0.5
0
0.5
1
1.5
2
2.5
3
Beat frequency (THz)
Figure 7.6: Terahertz radiation power versus beat frequency for an array with 5
elements. The thickness of the core layer is hf = 0.573 µm. The thickness of the
substrate at the frequency of 1 THz is , hs = 22.4 µm. The optical power of each
laser is 100 mW. The lengths of the array elements are 220 nm, 282 nm, 395 nm,
663 nm, and 2.814 µm. The thickness and the width of the array elements are 0.573
µm and 200 µm, respectively. The absorbed optical in each element is 39.5 mW.
The applied electric field over each element is 2.5 V/µm.
133
Chapter 8
Concluding Remarks
8.1
Summary and Contributions
The photomixing phenomenon in superconductors has been studied in detail. The
analytic expression for the generated photocurrent inside a superconductor film
excited by two CW laser beams has been derived using the fundamental carrier
transport equations for the superconductor material. The two-temperature model
has been used to find the spatiotemporal distribution of the electron and phonon
temperatures due to the interfering laser beams inside the superconductor film.
The resulting electron and phonon temperatures as well as the amplitude and the
phase of the generated photocurrent for an HTS photomixer have been analyzed
and their dependencies on the bath temperature, the beat frequency, and the angle
between the two laser beams have been explored.
The interaction of the laser and the photoconductor in a typical photomixing
scheme has been studied. The continuity equations for the carriers have been
solved in their general form for a photomixing configuration, in which two CW
134
laser beams excite a dc biased photoconductor. A general analytical expression for
the photocurrent has been derived. The dependencies of the generated terahertz
photocurrent on the physical parameters as well as the photomixer configuration
have been explored. The saturation behavior of the terahertz photocurrent with the
applied dc bias has been explored. The developed models can be used for designing
and optimization of terahertz sources.
A terahertz photoconductive photomixer-antenna device based on the integration
of a photomixer and an antenna element has been proposed. Two longitudinal
and transversal schemes, in which the optical illumination planes are respectively
parallel and normal to the direction of the applied bias, are investigated in detail.
The analytic expressions for the photocurrent distribution and the radiation power
have been derived for both configurations. The generated photocurrent inside the
photoconductor film acts as a radiation source and by choosing proper values for
the dimensions of the antenna configuration, the proposed structure radiates out
the terahertz power to the free space.
A continuous-wave terahertz photomixer array source made of LTG-GaAs has been
designed and analyzed. It has been shown that a few micro-watt terahertz power is
achievable from a typical array source at 1 THz. The dependency of the radiation
power on the beat frequency and the applied dc bias has been studied. It also
has been shown that the radiation beam can be steered over more than ±30o by
changing the angle between the two exiting laser beams.
A terahertz photomixer-antenna array source made of YBa2 Cu3 O7−δ thin film has
been designed and analyzed. It has been shown that a sub-milliwatt terahertz
power is achievable from a typical array structure. It also has been shown that the
radiation beam can be steered more than ±20o by changing the angle between the
two exiting laser beams.
135
Two terahertz photomixer-antenna array configurations with integrated optical excitation have been designed and simulation results are presented. In the proposed
structures, the laser beams are guided inside the substrate and excite the photomixer elements. It has been shown that a few nW terahertz power can be achieved
using two lasers with 100 mW output power.
The proposed devices are attractive for the terahertz applications, including terahertz spectroscopy and sensing, terahertz imaging, and local oscillators in terahertz
heterodyne receivers.
8.2
Future Work
Followings are some possible directions for the further research work:
• Fabrication and test of the proposed photomixer sources
• Looking for low-cost solutions for photomixer sources
One of the ideas can be using light emitting diodes (LEDs) instead of lasers
in photomixers to lower their complexity and cost. LEDs are cheaper than
lasers and do not need temperature and current controlling systems. The
challenging problems with LEDs are their wide line-width, phase noise, and
random output polarization.
Most of the developed photoconductive photomixers are made of LTG-GaAs
and use lasers with wavelengths from 750 nm to 850 nm. Some researchers
are working on new compositions suitable for photomixing at 1310 nm-1550
nm optical telecommunication wavelengths, for which lasers, detectors, and
variety of other optical components are commercially available at relatively
lower costs. The two most promising materials are InGaAs containing ErAs
136
particles and GaSb containing ErSb nanoparticles. Realization of a terahertz
photomixer working at the optical telecommunication wavelengths is a great
success from both technology and cost point of view.
Another low-cost solution can be looking at possibility of using polymers as
mixing media in photomixers. Polymers can be produced and implemented
relatively easier than high-quality photoconductors, and if one can use them
as mixing media in a photomixer, the overall cost will drastically decrease.
• Using electromagnetic band gap (EBG) structures in photomixers
EBG structures can be incorporated into the photomixer-antenna devices
to suppress substrate modes and increase their efficiencies. There are also
interesting reports on terahertz sources with guided schemes, which suffer
from high metallic loss in their waveguiding sections. Using EBGs as guiding
medium for generated terahertz signals can be a solution to this problem.
• Studying the effects of the lasers’ characteristics on the performance
of a photomixer
The frequency spectrum of the generated signal in a photomixer is a function
of the frequency spectrum and the phase noise of the exciting lasers. Using the
developed model for photomixers, one can study the effects of finite line-width
and phase noise of the exciting lasers on the performance of photomixers.
137
Appendix A
Calculation of the Carrier
Densities N0 and P0
To find N0 and P0 , one has to solve the nonlinear coupled equation set given by
Eqs. (4.24) and (4.25). The resulting equation for N0 is
κ1 N03 + κ2 N02 + κ3 N0 + κ4 = 0
(A.1)
where
κ1
κ2
κ3
κ4
i
h
e 1
−αd
(1 − e ) ηp µp τp + ηn µn τn
=−
ǫ αd
ηp µ p τp
e 1
(1 − e−αd )τn τp G0 −
−1
= (µn + µp )ηn ηp
ǫ αd
ηn µ n τn
³ 2η µ τ
´
p p p
=
+ τn ηn G0
µn
µp
= − τn τp ηn ηp G20
µn
138
(A.2)
(A.3)
(A.4)
(A.5)
with ηn = 1/(1 − α2 L2n ) and ηp = 1/(1 − α2 L2p ).
One can find P0 in terms of N0 as
P0 =
1
1
τn µn eǫ αd
(1
−
e−αd )
Ã
e
τn G0
τn µn N̄0 + ηn−1 −
ǫ
N0
!
(A.6)
An approximate solution to Eqs. (4.24) and (4.25) is
τn G0
≈ τn G 0
1 − α 2 τ n Dn
τp G0
P0 =
≈ τp G 0
1 − α 2 τ p Dp
N0 =
139
(A.7)
(A.8)
Publications
Journal articles
[1] D. Saeedkia, R. R. Mansour, and S. Safavi-Naeini “The interaction of laser
and photoconductor in a continuous-wave terahertz photomixer,” IEEE J.
Quantum Elec., Vol. 41, No. 9, pp. 1188-1196, September 2005.
[2] D. Saeedkia, R. R. Mansour, and S. Safavi-Naeini “Modeling and Analysis
of High-Temperature Superconductor terahertz Photomixers,” IEEE Trans.
Appl. Superconduct., Vol. 15, No. 3, September 2005.
[3] D. Saeedkia, R. R. Mansour, and S. Safavi-Naeini “A Sub-Milliwatt Terahertz
High-Temperature Superconductive Photomixer Array Source: Analysis and
Design,” IEEE Trans. Appl. Superconduct., Vol. 15, No. 4, September 2005.
[4] D. Saeedkia, A. H. Majedi, S. Safavi-Naeini, and R. R. Mansour, “Analysis
and design of photoconductive integrated photomixer/antenna for THz applications,” IEEE J. Quantum Elec., Vol. 41, No. 2, pp. 234-241, February
2005.
[5] D. Saeedkia, R. R. Mansour, and S. Safavi-Naeini “Analysis of a Photoconductive Photomixer Array Source Designed for Terahertz Applications,” IEEE
Trans. Antennas Propagat., in press.
[6] D. Saeedkia, A. H. Majedi, S. Safavi-Naeini, and R. R. Mansour, “HighTemperature Superconductive Photomixer Patch Antenna: Theory and Design,” Special Issue on Recent Progress in Microwave and Millimeter-wave
Photonics Technology, IEICE Trans. Electron., Vol. E86-C, No.7, pp. 13181327, July 2003.
140
[7] D. Saeedkia, A. H. Majedi, S. Safavi-Naeini, and R. R. Mansour, “Frequencyand Time-Varying Scattering Parameters of a Photo-Excited Superconducting
Microbridge,” IEEE Microwave Wireless Comp. Lett., Vol. 15, No. 8, pp.
510-512, August 2005.
[8] A. H. Majedi, D. Saeedkia, S. K. Chaudhuri, and S. Safavi-Naeini, “Physical
Modeling and Frequency Response Analysis of A High-Temperature Superconducting THz Photomixer,” IEEE Trans. Microwave Theory Tech., Vol.
52, No. 10, pp. 2430-2437, October 2004. Special Issue on Terahertz Electronics.
Refereed conference papers
[1] D. Saeedkia, A. H. Majedi, S. Safavi-Naeini, and R. R. Mansour, “CW Photoconductive Photomixer/Antenna Array Source for THz Applications,” IEEE
International Topical Meeting on Microwave Photonics, Ogunquit, Maine,
USA, October 2004. MWP’04 Technical Digest, October 2004, PP. 317-320.
[2] D. Saeedkia, A. H. Majedi, S. Safavi-Naeini, and R. R. Mansour, “Photomixing properties of high-temperature superconductors and photoconductors:
analysis and comparison,” presented at the SPIE International Symposium
Photonics North, September 2004, Ottawa, Canada. Optical Components and
Devices, Proceedings of SPIE, Volume 5577, PP. 648-658.
[3] D. Saeedkia, A. H. Majedi, S. Safavi-Naeini, and R. R. Mansour, “A CW Photoconductive Integrated Photomixer/Antenna THz Source,” SPIE Photonics
West Symposium 2004, San Jose, California, USA. Terahertz and Gigahertz
Electronics and Photonics III, Proceedings of SPIE, Volume 5354, PP. 94-103.
141
[4] D. Saeedkia, A. H. Majedi, S. Safavi-Naeini, and R. R. Mansour, “A novel
concept for integrated high-temperature superconductive photomixer patch
antenna,” International Superconducting Electronics Conference, 2003, Sydney, Australia.
[5] D. Saeedkia, S. Safavi-Naeini, and R. R. Mansour, “A Photoconductor Photomixer Array Source with Integrated Excitation,” Joint 30th International
Conference on Infrared and Millimeter Waves and 13th International Conference on Terahertz Electronics, Virginia, USA, September 2005, to be presented.
[6] D. Saeedkia, S. Safavi-Naeini, and R. R. Mansour, “A High-Temperature
Superconductor Photomixer/Antenna Array Source for Terahertz Applications,” Joint 30th International Conference on Infrared and Millimeter Waves
and 13th International Conference on Terahertz Electronics, Virginia, USA,
September 2005, to be presented.
[7] D. Saeedkia, S. Safavi-Naeini, and R. R. Mansour, “Modeling of High-Temperature
Superconductor Terahertz Photomixers,” Joint 30th International Conference
on Infrared and Millimeter Waves and 13th International Conference on Terahertz Electronics, Virginia, USA, September 2005, to be presented.
[8] D. Saeedkia, S. Safavi-Naeini, and R. R. Mansour, “Modeling of Photoconductor Terahertz Photomixers,” Joint 30th International Conference on Infrared and Millimeter Waves and 13th International Conference on Terahertz
Electronics, Virginia, USA, September 2005, to be presented.
142
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